Is the second postulate of Einstein's special relativity an axiom? I'm trying to grasp Einstein's special relativity theory and can't seem to find a clear answer as to why Einstein concluded that the speed of light is constant to observers in different inertial frames.


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*Sometimes I read it's because of the Michelson-Morley experiment. But that only proves there's no aether/medium for light to travel through.

*Sometimes I read that he derived it from Maxwell's equations. I still
need to study those, so I can't judge about that.

*And sometimes I read that it's just an axiom that we need to accept.
So what's the right answer?
 A: All of the above. Let's examine them in turn:

  
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*Sometimes I read it's because of the Michelson-Morley experiment. But that only proves there's no aether/medium for light to travel through.
  

This seems a slightly back-to-front way of looking at it. What the Michelson-Morley experiment shows is that the round-trip speed of light is constant in every direction and doesn't change as the Earth moves around the Sun. From this we conclude that the luminiferous ether theory is incorrect, since it predicts that the speed of light would be different depending on the Earth's motion relative to the ether. But the actual measurement essentially just says the speed of light doesn't depend on the reference frame, at least for observers fixed to the surface of the Earth.
(In fact one can't even really conclude that there is no ether, since the measurements could also be explained by the Earth dragging the ether around with it somehow. I believe this was seriously considered at the time. The ether theory is ruled out by the development of special and general relativity and the much better predictions they make, not by the Michelson-Morley experiment alone.)
It should be noted that Einstein himself said that the Michelson-Morley experiment was not a big part of his own personal motivation, as quoted in Grayscale's answer. Instead, his main motivation was to do with Maxwell's equations as described below. But clearly the idea of frame invariance was 'in the air' at the time, and both Michelson and Einstein were influenced by common ideas. (In fact, Lorentz had already worked out the mathematics of length contraction and time dilation, but interpreted them as physical phenomena rather than observer-dependent ones. So there were plenty of other people working on these kinds of ideas at that time as well.)


  
*Sometimes I read that it's derived from Maxwell's equations. I still need to study those, so I can't judge about that.
  

This is true too. The point is that if you assume Galilean invariance (i.e. if you don't assume relativity but you do assume that the laws of physics are the same for all observers) then it's possible to have an observer moving at the speed of light. If such an observer were to travel parallel to a light wave then they would see the electric and magnetic fields varying sinusoidally in space but not changing over time. But such a frozen light wave is not a solution to Maxwell's equations, so there is a paradox. 
One of Einstein's main motivations was to resolve this, i.e. to express Maxwell's equations in such a way that they would be valid for all inertial observers, instead of only being valid for an observer "at rest".


  
*And sometimes I read that it's just an axiom that we need to accept.
  

This is true too. The theory of special relativity is very mathematically elegant in that it requires very few assumptions. If you make this assumption (plus a few other quite minimal ones), then the rest of the theory follows, much as if you assume Euclid's axioms then you can derive Pythagoras' theorem.
If you're a pure mathematician then that's all you need to know - you don't care about why you should make the assumptions you make, so you call them axioms. But in physics we generally don't just make assumptions for no reason - they are made because of measurements or because of holes in existing theories. Einstein made the assumption of a constant speed of light for several reasons, but the Michelson-Morley experiment and the observer-dependence of Maxwell's equations were probably the main ones.
So all three are true. You can treat it as an axiom if you want, and that's probably the best approach if your main goal is to understand the maths. But the axiom also has several physical justifications, and that's what the other points you listed are.
A: None of the other answers seem to directly address the historical nature of the question, which asks why Einstein concluded that the speed of light is constant to observers in different inertial frames. Note this is clarified in the comments, where the asker states

"It's more like: hey, I'm Einstein, living in 1905 and I just came up
  with my second postulate. Here's why.... that's the answer I'm looking
  for."

So, to answer...
Among your three options, it is my understanding that the historical evidence suggests a mixture of the second and third options: Einstein had certain (axiomatic, if you like) convictions about absolute motion, and from the interplay of these convictions with (results derived from) Maxwell's equations, he was motivated to create his 1905 theory. Contrary to what some of the other answers suggest, Einstein was motivated much less by the Michelson-Morley result.
I think this quote from Einstein sums all this up fairly well:

In my own development Michelson’s result has not had a considerable
  influence. I even do not remember if I knew of it at all when I wrote
  my first paper on the subject (1905). The explanation is that I was,
  for general reasons, firmly convinced that there does not exist
  absolute motion and my problem was only how this could be reconciled
  with our knowledge of electro-dynamics. One can therefore understand
  why in my personal struggle Michelson’s experiment played no role or
  at least no decisive role.

(Emphasis added.)
However, the role that the Michelson-Morley experiment played in constributing to the ideas in Einstein's 1905 paper is a subject of debate. This paper discusses some of the conflicting lines of evidence as to how much Einstein knew about Michelson-Morley and how much it influenced his initial theory (note that the above quote is contained within the linked paper, as well as additional ones which may better expose some of the nuances of what contributed to Einstein's thinking).
Hope that helps!
A: I'll answer the third questions posed in the question in reverse order, and finally return to the third.


  
*And sometimes I read that it's just an axiom that we need to accept.
  

What's wrong with axioms? Einstein himself listed the constancy of the speed of light as a postulate (aka axiom; they're synonyms). Newton's laws of motion are axioms. Hilbert's sixth problem, announced in 1900, addressed the need to axiomatize physics. There's nothing wrong with axioms.
To the contrary; axioms in physics are a very good thing. They form the mathematical basis from which physicists predict the outcomes of experiments. One or more of the underlying assumptions (another name for axioms) are falsified if the outcome disagrees with the prediction.  


  
*Sometimes I read that he derived it from Maxwell's equations. I still
  need to study those, so I can't judge about that.
  

At the time Einstein developed special relativity, Maxwell's equations were consistent with experiments of an electromagnetic nature while Newtonian mechanics appeared to be inconsistent with such experiments. This was one of the key conundrums of late 19th century physics. One of those axiomatizations appears to be incorrect.
One of the many face-value consequences of Maxwell's equation is that the one-way speed of electromagnetic radiation (which includes light) is the same to all inertial observers. In particular, the expression for the speed of a wave of electromagnetic wave in vacuum depends on neither the speed of the source nor the receiver. It's simply $c = 1/\sqrt{\mu_0 \varepsilon_0}$, where $\mu_0$ and $\varepsilon_0$ are the magnetic permeability and electrical permittivity of space. Einstein simply took Maxwell's equations at face value, that the speed of light in vacuum is the same to all observers.

  
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*Sometimes I read it's because of the Michelson-Morley experiment. But that only proves there's no aether/medium for light to travel through.
  

The Michelson-Morley experiment disproves the existence of a luminiferous aether as conceived of at the time of that experiment. It does not disprove the existence of an ether. A competing axiomatization, Lorenz Ether Theory, developed at the same time Einstein developed his special theory of relativity, maintained the concept of a preferred frame of reference and of an ether. More on this later.
As Grayscale noted in his answer, the Michelson-Morley experiment had little if any influence on Einstein's thinking. Einstein was much more influenced by the Fizeau experiment, which looked at the speed of light in a moving medium such as flowing water. If Newtonian mechanics was true, such a medium should have a linear affect on the speed of electromagnetic waves, just as blowing air linearly effects the speed of sound waves. This was not what the Fizeau experiment found. It instead found a relationship that, after the fact, was found to be consistent with special relativity (and also Lorenz Ether Theory).


  
*And sometimes I read that it's just an axiom that we need to accept.
  

You don't need to accept the constancy of the one-way speed of light as axiomatic. You could accept Lorenz Ether Theory instead. It is indistinguishable from special relativity in terms of predicted experimental outcomes. The problem with Lorenz Ether Theory is that it is chock full of "where did that come from" (WTF) assumptions. It makes length contraction, time dilation, a preferred frame of reference, and an ether that acts as a medium for electromagnetic radiation in empty space axiomatic. Length contraction and time dilation nicely hide the preferred frame of reference and the ether from any experimental test designed to detect them. Nice. (That was a sarcastic "nice".)
Special relativity only has one "where did that come from" assumption, the constancy of the one-way speed of light, and that is only a ""where did that come from" assumption because we are too preconditioned by Newtonian thinking. The one-way constancy of the speed of light is a beautiful thing that speaks directly about the geometry of space-time. In contrast, Lorenz Ether Theory is a rather ugly thing that fails Occam's razor that does not speak directly to the mathematics that underlies space and time.
A: Nathaniel's answer accurately addresses your question. I would like to address your specific doubts.

  
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*Sometimes I read it's because of the Michelson-Morley experiment. But that only proves there's no aether/medium for light to travel through.
  

True, and this in turn implies that the velocity of light must not be measured with respect to the aether. On the contrary, if $c$ has to have an absolute meaning in the Maxwell's equations, then it has to have an absolute value, i.e. it must be the same in all inertial frames.


  
*Sometimes I read that it's derived from Maxwell's equations. I still need to study those, so I can't judge about that.
  

The important point here is that it can be derived from Maxwell's equations provided that you know the correct transformation laws between inertial frames. In turn, Einstein derived these laws by postulating the constancy of $c$. But you could still postulate the transformation laws, and derive the constancy of $c$ through these laws or through Maxwell's equations. Or you could postulate (as Lorentz did) that Maxwell's equations should be the same in all inertial systems and use the postulate to derive the transformation laws.


  
*And sometimes I read that it's just an axiom that we need to accept.
  

This is usually what you do.
A: You need to be very careful as to what you mean by "the speed of light". There are indeed one-way speeds and two-way speeds and they have completely different a status.


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*One-way. A light signal departs from $A$ at time $t_A$ and arrives at $B$ at time $t_B$, as marked by clocks respectively located at $A$ and $B$. The one-way speed is then $AB/(t_B - t_A)$.

*Two-way. A light signal departs from $A$ at time $t_A$, hits a mirror at $B$ and is reflected back to $A$ where it arrives at time $t'_A$, where those times are recorded by a clock sitting at $A$. The two-way speed is then $2AB/(t'_A-t_A)$.


The one-way speed of light is a convention as it depends on how the clocks at $A$ and $B$ are synchronised. The two-way speed of light, on the contrary, is a genuine physical observable as it can be measured. In his 1905 paper, Einstein chose a synchronisation procedure for the clocks such that the time of flight of a light signal from $A$ to $B$ is equal to the time of flight from $B$ to $A$. As a result the one-way speed from $A$ to $B$ is equal to the one-way speed from $B$ to $A$, and therefore also to the two-way speed.
Your question is therefore ill-posed. If you meant the one-way speed, then being a convention, we can't say more without choosing one such convention. If we go for Einstein's method, then it boils down to discuss the two-way speed. But we could choose anything else.
If you meant the two-way speed, then Michelson-Morley experiment shows that it is isotropic at the surface of Earth. Now consider the most popular Ether theory of that time past: the Ether is at rest in the geocentric frame, or even in the heliocentric frame. Then the M&M apparatus, thanks to the Earth spinning and revolving about the Sun moves with a great variety of speeds with respect to the Ether. So we can say that the isotropy of the two-way speed of light is observed in the corresponding great variety of inertial frames. Thus assuming that it is isotropic in any inertial frame is a legit assumption to make, to see where that leads when one tries to find space and time transformations between frames which would realise that. That's precisely what Einstein did in his 1905 paper.
A: Einstein "borrowed" his constant-speed-of-light postulate from the ether theory:
Albert Einstein: "...I introduced the principle of the constancy of the velocity of light, which I borrowed from H. A. Lorentz's theory of the stationary luminiferous ether..."  https://en.wikipedia.org/wiki/Lorentz_ether_theory
In 1887 the Michelson-Morley experiment UNEQUIVOCALLY confirmed the variable speed of light posited by Newton's emission theory of light and refuted the constant (independent of the speed of the light source) speed of light posited by the ether theory and later adopted by Einstein as his 1905 second postulate:
John Norton: "The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."  http://philsci-archive.pitt.edu/1743/2/Norton.pdf
Banesh Hoffmann, Relativity and Its Roots, p.92: "Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."  https://www.amazon.com/Relativity-Its-Roots-Banesh-Hoffmann/dp/0486406768
