Why is there a controversy on whether mass increases with speed? Some people say that mass increases with speed while others say that the mass of an object is independent of its speed.
I understand how some (though not many) things in physics are a matter of interpretation based on one's definitions. But I can't get my head around how both can be "true" is any sense of the word.
Either mass increases or it doesn't, right?
Can't we just measure it, and find out which "interpretation" is right? E.g. by heating up some particles in a box (in some sophisticated way) and measuring their weight?

UPDATE:
Right, so I've got two identical containers, with identical amounts of water, placed on identical weighing scales, in the same gravitational field. If one container has hotter water, will the reading on its scale be larger than the other? If the answer is yes, and $g$ is constant, does this mean that the $m$ in $W=mg$ has increased?
 A: 
Some people say that 'mass increases with speed'. Some people say that
  the mass of an object is independent of its speed. I understand how
  some things in physics are a matter of ... definitions. But, I can't
  get my head around how both be 'true' is any sense of the word. 
  Either mass increases or it doesn't, right?
  Can't we just measure it... heat up some particles in a
  box and measure their weight.

The technicalities of the issue have been masterly presented. I'll try to give you a more simple 'user-friendly' explanation.You make some confusion in your own post, between mass and weight, and if/when you clarify that it can help you bring correctly into focus the problem.
Suppose you can count literally up the (electrons/protons atoms) of your body considering as an average an atom of carbon 12. That number is dimensionless, absolute (instead of weighing it, which is relative). Suppose you ascertain that the mass of your body is made up by $10^{27}$ atoms. That mass is the real mass of your body and it can/will never increase.
Now, suppose you weigh your body on the Earth then on the Moon and then on Jupiter, what do you get? that your 'mass' apparently increases and decreases. You seem to have accepted that, forgetting that your body sill has the same number of atoms.
You have accepted so far that the same mass can be 'observed' to have different values in different circumstances, in this case: gravity.
Now, try to apply the same logical mechanism that made you accept this apparent contradiction to another situation in which what varies is speed: when a body acquires kinetic energy it acquires (temporarily, as long as it conserves that KE) the same property that your body acquired on Jupiter. Your body at 0.8 c weighs much more than when it is travelling at 0.01 c, yet its 'true mass' is still made up by $10^{27}$ atoms.
In this case, besides gravity, you might find a more simple, 'rational', explanation that can make it easier for you to understand and to accept it: energy (kinetic, thermal etc) bound in a body has a tiny 'mass/weight' attached to it, which temporarily increases its 'weight'

Can't we just measure it... heat up some particles in a
  box and measure their weight.

It is not clear what you are trying to prove with that, but if you heat matter its weight will change, due also to 'gravity'

If you have absolutely identical objects that have the same weight
  exactly when they are at the same temperature, then when one object is
  heated, it will weigh more. This is because the gravitational force
  depends on the stress energy tensor in general relativity. The stress
  energy tensor 00 component is the total energy of the body, which
  includes the rest mass plus the kinetic energy of the object.
  Temperature differences means that there is a different amount of
  kinetic energy in the motion of the atoms of the two bodies.
For example, if you start with two identical kilograms of water at 0
  Celsius, and if you then heat one of them to 100 Celsius, then the
  kilogram at 100 Celsius would be heavier by an amount equivalent to
  4.6 nanograms of additional water weight (see 100*1000 calories / c^2 ).
Now 4.6 nanograms is not very much, but it is equivalent to 154
  trillion molecules of water (see 4.6 10^-9 gm water in molecules ).
  Just imagine - the energy used to heat the water is equivalent to the
  weight of 154 trillion additional water molecules if they could be
  converted completely into energy (remember E=mc^2)!

This extra mass/weigth is temporarily added to your body, and when it slows down or cools off, it  loses energy and consequently its 'weight' attached to it and returns to its 'true' value. Does this help you clarify your doubts?
A: There is nothing wrong or awkward with defining relativistic mass, and it's not obselote. Physicists do refer to relativistic mass all the time, they just call it "energy". Relativistic mass is to proper mass what co-ordinate time is to proper time -- that's all there is, and relativistic mass is as obselote as clocks are.
A: If ever you find yourself wondering about how to consider the speed-dependent effects of relativity upon a body- namely time dilation, length contraction and mass increase- you might find it helpful to ground your thoughts by recalling that there is no such thing as an absolute speed. All speed is relative, which means that an object never has a unique speed, but instead it simultaneously has every possible speed relative to every possible reference frame.
Suppose you have a kilogram weight in your hand. Its intrinsic mass is a kilogram. As you hold the object stationary in your hand, it is moving at 0.5c relative to some reference frame, at 0.00001c relative to another, at 0.9999999c relative to a third, and so on endlessly. In each of those frames the effective mass of the object- ie the ratio between forces applied to it and its resulting acceleration- will have a different value. Clearly the intrinsic mass of the object in your hand cannot have different values simultaneously, so it must remain independent of reference frame. Its effective mass is dependent on reference frame, but its intrinsic mass is constant.
The same holds true for length contraction. A metre rule in your hand is simultaneously moving at arbitrary possible speeds in other reference frames in which its contracted length can take any value less than a metre. Clearly it cannot have an infinite number of intrinsic lengths simultaneously, so length contraction is an effect that exists only in reference frames moving relative to the metre rule- the ruler itself does not contract.
Likewise with time dilation. Your heart is beating at a certain rate. When viewed from a reference frame moving at close to the speed of light, your heart might appear to beat only once a year. Clearly the intrinsic rate at which your heart beats is does not change.
All of the effects of relativity are 'real' in the sense that they can be observed and measured, but they are only 'real' in reference frames moving relative to the object that appears to be subject to the effect. The object itself, intrinsically, does not change. Moreover, all of the effects arise directly from the fact that reference frames moving relative to each other do not have common planes of simultaneity- instead their time axes are tilted relative to each other. It is the fact that in different frames they view an object at different combinations of time that causes the observed effects.
A: As in Ben Crowell's Answer, the concept of "Relativistic Mass" is not wrong, but it is awkward. There are several things a loose usage of the word "mass" could imply, all different and thus it becomes a strong convention to talk about the meaning of the word "mass" that is Lorentz invariant - namely the rest mass, which is the square Minkowski "norm" of the momentum 4-vector. Given its invariance, you don't have to specify too much to specify it fully, and so it's the least likely one to beget confusion.
Here's a glimpse of the confusion that might arise from the usage of the word mass. To most physicists when they learn this stuff, the first time they see "mass" they think of the constant in Newton's second law. So, what's wrong with broadening this definition? Can't we define define mass as the constant linking an acceleration with a force? You can, but it depends on the angle between the force and the velocity! The body's "inertia" is higher if you try to shove it along the direction of its motion than when you try to introduce a transverse acceleration. Along the body's motion, the relevant constant is $f_z=\gamma^3\,m_0\,a_z$, where $m_0$ is the rest mass, $f_z$ the component of the force along the body's motion and $a_z$ the acceleration begotten by this force. At right angles to the motion, however, the "inertia" becomes $\gamma\,m_0$ (the term called relativistic mass in older literature), i.e. we have $f_x=\gamma\,m_0\,a_x$ and $f_y=\gamma\,m_0\,a_y$. In the very early days people spoke of "transverse mass" $\gamma\,m_0$ and "longitudinal mass" $\gamma^3\,m_0$. Next, we could define it as the constant relating momentum and velocity. As in Ben's answer, we'd get $\gamma\,m_0$. We can calculate $\vec{f}=\mathrm{d}_t\,(\gamma\,m_0\,v)$ correctly, but not $\vec{f}=\gamma\,m_0\,\vec{a}$, it fails not only because $\gamma$ is variable but also because the "inertia" depends on the direction between the force and velocity.
So, in summary, "inertia" (resistance to change of motion state by forces) indeed changes with relative speed. You can describe this phenomenon with relativistic mass, but it is awkward, complicated particularly by the fact that the "inertia" depends on the angle between the force and motion. It is much less messy to describe dynamical phenomena  Lorentz covariantlt, i.e. through relating four-forces and four-momentums and one uses the Lorentz invariant rest mass to see these calculations through.
A: There is no controversy or ambiguity. It is possible to define mass in two different ways, but: (1) the choice of definition doesn't change anything about predictions of the results of experiment, and (2) the definition has been standardized for about 50 years. All relativists today use invariant mass. If you encounter a treatment of relativity that discusses variation in mass with velocity, then it's not wrong in the sense of making wrong predictions, but it's 50 years out of date.
As an example, the momentum of a massive particle is given according to the invariant mass definition as
$$ p=m\gamma v,$$
where $m$ is a fixed property of the particle not depending on velocity. In a book from the Roosevelt administration, you might find, for one-dimensional motion,
$$ p=mv,$$
where $m=\gamma m_0$, and $m_0$ is the invariant quantity that we today refer to just as mass. Both equations give the same result for the momentum.
Although the definition of "mass" as invariant mass has been universal among professional relativists for many decades, the modern usage was very slow to filter its way into the survey textbooks used by high school and freshman physics courses. These books are written by people who aren't specialists in every field they write about, so often when the authors write about a topic outside their area of expertise, they parrot whatever treatment they learned when they were students.  A survey [Oas 2005] finds that from about 1970 to 2005, most "introductory and modern physics textbooks" went from using relativistic mass to using invariant mass (fig. 2). Relativistic mass is still extremely common in popularizations, however (fig. 4). Some further discussion of the history is given in [Okun 1989].
Oas doesn't specifically address the question of whether relativistic mass is commonly used anymore by texts meant for an upper-division undergraduate course in special relativity. I got interested enough in this question to try to figure out the answer. Digging around on various universities' web sites, I found that quite a few schools are still using old books. MIT is still using French (1968), and some other schools are also still using 20th-century books like Rindler or Taylor and Wheeler. Some 21st-century books that people seem to be talking about are Helliwell, Woodhouse, Hartle, Steane, and Tsamparlis. Of these, Steane, Tsamparlis, and Helliwell come out strongly against relativistic mass. (Tsamparlis appropriates the term "relativistic mass" to mean the invariant mass, and advocates abandoning the "misleading" term "rest mass.") Woodhouse sits on the fence, using the terms "rest mass" and "inertial mass" for the invariant and frame-dependent quantities, but never defining "mass." I haven't found out yet what Hartle does. But anyway from this unscientific sample, it looks like invariant mass has almost completely taken over in books written at this level.
Oas, "On the Abuse and Use of Relativistic Mass," 2005, here.
Okun, "The concept of mass," 1989, here.
A: There's no controversy about whether mass increases or not, there's controversy about what you call mass. One possible definition is that you consider some object's rest frame, and call the $\tfrac{F}{a}$ you measure there (for small accelerations) the mass. This notion of mass can't change with speed because, by definition, it's always measured in a frame where the speed is zero.
There's nothing wrong about this way of thinking, it's basically a question of mathematical axiom. Only, it's not really useful to require the rest frame, because we're constantly dealing with moving objects1. Therefore, the (I believe) more mainstream opinion is that that quantity should only be called rest mass $m_0$. The actual ("dynamic") mass is defined by what we can directly measure on moving objects, and, again simply going by Newtons law, if you e.g. observe an electron moving with an electric field at $0.8\:\mathrm{c}$, you'll notice it is accelerated not with $a = \tfrac{F}{m_0}$ but significantly slower, namely as fast as a nonrelativistic electron with mass $m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$ would. It is therefore reasonably to say this is the actual mass of the electron, as seen from laboratory frame.

1 Indeed, you can argue it's never possible to really enter the rest frame. In macroscopic objects you'll have thermal motion you can't track, and yet more fundamentally there's always quantum fluctuations.

Edit as noted in the comments, amongst physicists there will of course not really be controversy about what mass definition is meant: they'll properly specify theirs, usually just following the convention of invariant mass. That can easily be calculated for any given system, from the total energy and momentum rather than the actual movements of components (which, again, you can't track). That still leaves scope for confusion to the unacquainted though, because whether the invariant mass increases or not when accelerating an object depends on whether you consider the mass of some bigger system, say with some much heavier stationary target, or the accelerated object on its own. This may seem counterintuitive, so when hearing accounts of the same experiment based on either of these "system" definitions you think there's a controversy, when really the accounts are just talking about different things.
A: Because unfortunately it gained the label "relativistic mass", which gave it a sort of unconscious legitimacy.
I propose we consider calling it instead "directional mass".  This is IMO much less likely to be taken seriously as a concept.
I intend to do it myself from now on, and see how it goes.
A: I think it is a question of the reference frame. You pick a reference frame tied to your object (the rest frame) then the mass is always the same in that frame and it is the rest mass of the object, $m_0$. 
If you pick another reference frame in which your object can move, its mass will be different and will definitely depend on its velocity. Its expression will be $m=\gamma m_0$, where $\gamma=1/\sqrt{1-v^2/c^2}$. 
The reason for this is that the energy of the moving object will be seen from your reference frame as kinetic plus the rest energy of that object. The total energy of the object is still $E=m c^2$, only this time $m$ will depend on the velocity of the object in the reference frame you chose.
In my limited experience with relativity theory (special and general), it appears that most of the confusion in understanding its workings come from not understanding the role of reference frame. Whenever you want to calculate anything you must first set the reference frame (a ruler, a watch and origins for both space and time axes). Once you do that you can make statements about the system you study. 
Sometimes you may have 2 objects moving relative to each other. Usually you can compute everything about those isolate objects much easier in their respective rest frames. Then you must worry about the whole system and you need to set a common reference frame for the system of two objects and calculate whatever you need to calculate (distances, velocities, electromagnetic fields) in that frame. For this, you need to use the transformations (Lorentz or Poincaré transformations, for example) to transform the quantities you calculated in those objects' rest frames to the common reference frame. 
A: So you have a body travelling at such a speed that it's $\gamma$ factor is non-negligible. How is the mass going to become manifest anyway? One way would be through its gravity: there could in principle exist a balance so big that it's set up in a uniform gravitational field, and the moving body is on one pan, and there is an identical stationary body on the other, and the pans are so big that despite the near-luminal speed of the moving body we have time to see how much mass needs to be added alongside the stationary one to balance the balance; or maybe we could have two bodies initially coalesced, and two (two to obviate wobble) test bodies orbiting about them, and then they are propelled apart symmetrically at near-luminal speed, and the mass inferred from the change in the motion of the test bodies: or maybe we could infer the mass from momentum by colliding the moving body with something, or by some how measuring the reaction in whatever it was that propelled them: or through energy in a manner similar to that whereby we might reach it through momentum (particles created in a collision in an accelerator, that sort of thing): or ... 
Yes I realise the foregoing is a highly laboured disquisition; and also that much of it could not actually be performed by human: but it is intentionally laboured, as I am trying to demonstrate that beyond classical dynamics, mass is essentially inseperable from the way in which it is manifested. You can manifest it through momentum, in which case the immediate datum is the behaviour of momentum in the relativistic regime, or similarly through energy; or you can manifest it through gravity, in which case the immediate datum is the behaviour of objects under gravity when they are moving at near-luminal speeds. In no case is mass itself the immediate datum. (I am being a bit pedantic here, BTW, and using the word 'immediate' perfectly literally to mean without mediation rather than right now). Mass as immediate datum is really a classical concept, proceeding from being able to actually have the stuff there, examinable & palpable, and which has installed itself as a habit of thought that we could do with breaking when we move onto the relativity paradigm, as by the very stipulating of the moving frame of reference we are setting mass per se out of reach.
But then I'm considering motion in a straight line of course. I cannot find any straight answer to the question of whether a particle moving in a circle has a Lorentz factor on its mass. I think  in that case the entire system of kinetic energy of particle + its potential energy in whatever field is constraining it to circular motion must be considered as a whole. But we know that in nuclear reactions, the difference in mass of the substance from ante -reaction to post -reaction is equal to (energy emitted)÷c²: that is very thoroughly established experimentally; and we are talking about mass considered in an altogether unextraordinary sense - weighable in a balance. Both the potential energy of the binding of the particles together, and the kinetic energy of the motion of the particles around each other are making contributions to the total mass. And some of that potential & kinetic energy  can be extracted as energy in the usual sense; and the total mass is mass in the usual sense. So it would seem from this that maybe relativistic mass is not altogether merely an expedient abstraction. 
So a body moving in a straight line at relativistically significant speed is not practically weighable, although fabulous and yet physically realistic - though not humanly implementable - means can just be devised for (mediately) weighing a body moving at relativistically significant constant velocity in a straight line; whereas subatomic particles moving at high speed within a piece of matter constitute a system in which the contributions to the total mass of the kinetic energy of their motion and the potential energy of their being bound together cannot simply be extricated from each other in the relativistic régime ... although the system as whole may well be simply weighable. 
Looking for particularisations of this constrained non-linear relativistic motion, the only two I can find are essentially also quantum mechanical: the first one - the atomic one, is that of motion of electrons in an atom. This is dealt with by the rather difficult & complicated Dirac equation, although approximate results can be treated by setting relativistic effects as a perturbation. The perturbation approach is reasonable inthat for a hydrogen atom the motion is only barely relativistic - the  $\beta$ factor having a value equal to the fine structure constant $\alpha$. Whichever method you take, the result is an expansion in terms of $\alpha Z$, $Z$ being atomic number. Or $(\alpha Z)^2$, really, as the terms tend to be even. According to this, relativistic effects become the more significant the higher up the periodic table, and account for certain anomalies that set in, such as gold being yellow, and cæsium being more electropositive than francium; and also for the possibility that an atom cannot even exist with $Z>137$, as the mentioned series would diverge for such an atom. (Some, however, say that this effect might be circumvented, and that this argument does not in fact preclude the existence of such atoms.)
But the point here is, that in treating of an atom thus, there ceases to be any equivalent of the gorgeous virial theorem, whereby the total energy is equal to the negative of the kinetic energy and the potential energy is twice the total energy - this kind of segregation sloughs away, and it's all framed in terms of total energy alone. So the relativistic mass of the electron, & it's corresponding energy, becomes thoroughly conflated with the total energy of the system. The other system which is that of the quarks constituting baryons: the motion here is certainly ultrarelativistic but there is certainly no more talk about the 'shape' of the quark's 'path'! All that is longsince gone! You might well read that the restmass of the quarks (and you can't even have free quarks anyway! whence there is controversy over even defining the restmass of quarks atall! ) is a small(ish) fraction of the mass of the baryon, and that it's mass is very much preponderantly due to the energy of the quarks' binding & motion. So here we have relativistic mass being very palpable - but alongwith a thorough foiling of any hope of attributing it to a nice Lorentz factor with a nice $\beta$ & $\gamma$, & allthat.
Is the motion of particles in a cyclotron or synchrotron a sufficiently tight circle for the departure from the approximation of linear motion to be significant in this connection!?
I do see how the notion of relativistic mass is a very heavily fraught one - but on the other hand it does not seem altogether to be one that can be utterly dismissed.
