Does information have mass? Does information itself have any detectable mass? If so, how is the mass of information measured if at all possible? Mathematically, is it possible for information itself to have mass? What would be the equation to ask this question or to demonstrate it? Is there a practical, physical way to measure if information itself has mass, how can this measurement be achieved? Furthermore, if information does have mass, isn't it necessary for information to contain energy as well as per the general theory of relativity?
 A: Yes. Indirectly.
First, what is information? It is the ability to do prediction.
Second, entropy $S$  (dimensionless) is a measure of unavailable information, (for more see this answer), and we know entropy is proportional to energy. So, information and energy should also be related -  Landauer's principle (bits).
$k_BT[{\rm J}]\cdot \ln2[{\rm bits}]\le E$
Now, you might consider that at rest $E=mc^2$ and so 'deduce the mass' of a bit of information like Vopson, or simply recognise that (nats)
$E=k_BT[{\rm J}]\cdot S[\rm{nats}]$
Entropy (can be considered to be) dimensionless, mass-energy isn't. This all expands Luaans answer. Also, this is the 2010 experiment which demonstrates work can be extracted from information.
A: For an experimental physicist, as I am, the question sounds like comparing apples and oranges.
Why do we have basic units? So that we can measure and compare apples with apples and oranges with oranges.
What are the units of information?

Does information have mass?

There are various unit systems  for mass, in addition to kilograms.
For information there is a so called natural information unit, according to the wiki article

The natural unit of information (symbol: nat),sometimes also nit or nepit, is a unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the bit. This unit is also known by its unit symbol, the nat. The nat is the coherent unit for information entropy. The International System of Units, by assigning the same units (joule per kelvin) both to heat capacity and to thermodynamic entropy implicitly treats information entropy as a quantity of dimension one, with 1 nat = 1. Physical systems of natural units that normalize the Boltzmann constant to 1 are effectively measuring thermodynamic entropy in nats.

If a physical variable may be said to be related to information entropy it is energy per kelvin, but the relationship  just allows to see thermodynamic entropy in units of nats.
So, imo, information entropy and mass have no connection.
A: Information does not have mass, but the physical materials containing that information may.  For example, if one wishes to have robust storage of that information, one may choose a structure which represents that information in a way which has a considerable amount of energy.  That energy has mass by relativistic principles.  However, it is not the information storage that causes the mass, but merely the physical properties of a medium.
As a concrete example, consider an object which can encode information (such as a magnetic tape).  We have two identical tapes.  Onto one tape, we load random noise, with no information content.  Onto the other tape, we load information encrypted with a symmetric encryption key (we encrypt it here because one principle of encryption is that the encrypted results are indistinguishable from random bits unless you have the decryption key, making the comparison more clear).  The energy and mass of these tapes will be identical, while one carries information and the other does not.  The only way to distinguish them is to read their contents and decrypt the data.
As a more extreme step, consider what would happen if we destroyed the key.  If information did indeed have mass, the second tape would be obliged to lose mass when we destroy the key.
A: Information does not have mass. Photons carry information and they are massless. One could ask if data has mass but that is also no.
A: Claude Shannon proposed the idea of information entropy, which is essentially about how much uncertainty you have about different outcomes. For example, when I read 100 bytes from a hard drive, I would (almost always) expect to get the same 100 bytes from the hard drive over and over again; if I read 100 bytes from a random number generator over and over, I would (almost always) expect to get different 100-byte sequence each time. In the first case, there is information - in the second, there is no information.
As it turns out, thermodynamic entropy is a kind of information entropy. I'm not going to dwell much on that because even giving a good explanation of thermodynamic entropy is tricky. But one can imagine a scenario where you can convert information into free energy. Consider two pistons, opposite in one chamber. There is a single molecule of "working fluid" between the two, and a removable partition. If you know which side of the partition the molecule is in, you can open and close the partition accordingly and produce useful work. In fact, this has been demonstrated in an experiment (though obviously not in any practical way). If you want to know more, have a look at Szilard's Engine. Note that what we have done was convert information to energy (regardless of how efficient the process actually is!).
Does this qualify as "information has energy"? Some claim it does, some don't. It's definitely weird to think about :)
A: Yes
The mass of information can be inferred from the Bekenstein Bound.  However, it depends on the spatial extent of the information: a larger space requires less mass per bit.  But don't worry...information is very "light": we can store up to $10^{43}$ bits per kilogram within a sphere of 1 meter radius.
Note that if the universe is really a simulation running on "God's computer",  Bekenstein gives us a lower bound on its hardware specs. ;)
Computation
If we consider information in terms of irreversible computation (the usual kind done by computers), then we must also account for the Landauer Limit.  This one is more difficult, because we can't tie it directly to mass.  It implies an energy expenditure to "perform" the computation, but really, it is only an entropy expenditure which can apparently be "paid for" by non-energy conserved quantities, like angular momentum (spin).
A more direct bound is provided by the Margolus-Levitin Theorem.  This puts an upper bound of $10^{33}$ on the number of "operations" per second per Joule of energy, which we can think of as the limit of "producing" new information via computation.  Via mass-energy equivalence, we could also state this bound as kind of "operations per second per kilogram" limit, which indirectly implies that computations also have mass.
*** EDIT ***
Encoded Information
Mr. Anderson's answer gives a very nice link (please upvote for this alone) to a paper by Dr. Vopson which describes the process of encoding a bit onto a storage medium.  This is perhaps the most natural and intuitive notion of "information" with which most people are familiar.  Vopson argues that the fact that the state persists without further energy input is due to the fact that the system actually increases in mass like so:

In this paper a radical idea is proposed, in which the process of holding information indefinitely without energy dissipation can be explained by the fact that once a bit of information is created, it acquires a finite mass, mbit. This is the equivalent mass of the excess energy created in the process of lowering the information entropy when a bit of information is erased.

The increase is mass depends on the temperature of the system, but he claims that:

...at room temperature (T = 300K), the estimated mass of a bit is ∼ $3.19×10^{-38} kg$.

He then goes on to propose an experiment whereby a 1 TB storage device is erased and then written to, magnifying this tiny mass by about $1\times10^{12}$.  Unfortunately, this only brings the "informational mass" into the range of $10^{-25}$ kg, which is roughly the weight of 60 H atoms.
Embodied Information
However, I will argue that the Bekenstein Bound referenced above is not about this so-called "encoded information", but rather refers to information which is intrinsic to a physical system.  More precisely, I believe it refers to the amount of information required to clone the quantum system, if quantum cloning were possible.  The bound explicitly defines an entropy, which is proportional to the number of microstates of the system.
So what does it mean for information to "have mass"?  The Bekensteinian version is perhaps disappointing compared to the Vopsonian version.  My interpretation is that for the universe to have information, it must have energy.  I presume that an empty universe with no energy also has no information.  But this also applies to a subregion of the universe as well.  Information can only exist in a region which also contains energy.  Furthermore, that energy has properties by which we can describe it.  It has degrees of freedom.  And those degrees of freedom result in an ensemble of possible microstates for that quantity of energy.  The information embodied by the energy simply encodes which microstate corresponds to the quantity of energy.
Thus, a photon flying through space might be encoding information, if, for instance, a human selected it amongst a population of photons because of one or more of its properties.  But regardless of the encoding, it also embodies information about its frequency, polarization, direction, etc.
More importantly, I will claim that all encoded information ultimately derives from embodied information via a selection process whereby particular microstates are chosen to represent information and other microstates are designated as "noise" and systematically filtered out or suppressed.
So ultimately, the idea that "information has mass" just boils down to the fact that information requires energy to exist, and mass and energy are equivalent.  Boring, huh?  The trick is that information does not have a fixed mass, but depends on energy density and particle number.
A: Information itself does not have mass.
A simple example here is having a series of coins, where I lay them down heads/tails based on a binary 1/0 state. I am able to essentially convey any information I want (that we are currently able to express electronically). I could describe the entire content of Wikipedia if I had enough coins.
But mass-wise, this is no different than if I had a bunch of coins without arranging them based on some binary logic.
That being said, there are ways to cheat the system in your favor here. If, for example, I would place the coins representing the binary 1 on a raised platform, then those coins have more potential energy, and you could argue that this information has energy, which in turn means it can have mass (good old E=MC²).
It's a stretch, but it's technically correct.
But then again, I could also arrange these coins on varying platforms without conveying any information, so the existence of the information still doesn't really force the amount of mass/energy contained in the system to be any different.

However, we as humans are pretty much unable to perceive anything massless, so you can argue that information, which is inherently intended for humans to perceive, indirectly requires something that has mass.
That's a different question from what you asked, though.

Also, just as a though experiment, if you consider that information could have mass, have you then also considered that it could have negative mass?
I'm thinking of carving a message into a stone tablet here. You are effectively removing bits of stone, thus lowering the mass rather than increasing it. If were were to hypothetically conclude that information has mass, it would seem contradictory that having the information contained on a stone tablet would then lower its mass.
A: No. As a trivial counterexample, take any system that has two different states of the same energy.
A: At the crux of the question is another question:
Can information exist without a medium for it to be held in/on as well as a means for interpreting what is in/on the medium?
I'm going to say no. You may have a different opinion.
Must the medium have mass?
Probably yes. In the case of bare photons, maybe it's debatable.
Must the means for interpreting have mass?
Well, interpreting something implies a "who" or "what" to do the interpreting. So, some system with some complexity must exist. I'm going to make the bold claim that such a system almost surely has mass.
The final question:
Is the concept of information separate from the means which makes it exist?
This is a pretty deep question and I haven't read nearly enough philosophy textbooks to attempt an answer.
Anyway, my point is: the answer to "Does information have mass?" is going to entirely depend on how you answer all the questions above, and I don't think these are answerable without opinion (especially not the final one).
A: It costs energy to create information, and therefore the energy of ``information creation'' has a mass.
Reading information also costs energy, and therefore has a mass associated with it.
But I don't think there are any current models that explicitly require that the information that was encoded with energy (and therefore mass) itself has mass.  You can certainly encode information on the frequencies/amplitudes/polarization of photons. While it costs energy to send them out into space, they certainly will exist without any masss while they propagate in space.
(Also, one thing to point out is that the creation of all information is NOT necessarily the same energy. For example, if I lay out a bunch of colored M&Ms on a table inside a van while its driving and the M&Ms are bumping around. It's going to be MUCH harder for me to keep all of the M&Ms separated in color than it is for them to be mixed up. The energy it requires to keep things from moving towards their natural most-likely "mixed-up" state is related to the "information entropy" - and the more likely it is for things to get mixed-up, the more energy it will cost to keep them in a particular state.)
A: Yes information does have mass because if it did not, then that means information can exist without mass but if there is no mass then there is no information so the only other option is information does have mass and you can measure it and the measurements exists today! computer bits, computer bit measure how much how information something stores for example one kilobyte is roughly how much information that's in half a novel.
