As per the Wikipedia entry for “Speed of Light”:
Apparently the top speed of light is a much more general concept than the name alludes to, because this same top speed also applies to phenomena such as information and gravity (theoretically). Why is light so fundamental out of all of these phenomena that it deserves a monopoly on the name for the universal speed limit?
The speed of light was measured before we figured out that light goes the fastest that anything can go.
Because Einstein was thinking about light when he first figured this out (i.e., that light must travel as fast as anything can possibly go). Also, at the time, it was possible to actually measure the speed of light, but not (at the time, I think) the other forms of electromagnetic radiation to which the same theoretical considerations also apply.
Information is only limited to light speed because it has to be carried by something physical, so it cannot move any faster than the fastest possible physical thing (i.e., light or other electromagnetic waves). The limitation on the speed of information is not an independent issue.
As a physicist, the first thought would be: maybe “all of these phenomena” are really just different manifestations of the same thing. The Holy Grail of physics is the Theory of Everything. One simple theory that explains everything. The universe needn’t be elegantly simple, but it sure seems that the more we learn, the more it all ties together.
Light is the only massless particle we can easily measure. That is all. All massless particles move at speed c.
What are the other massless particles besides the photon?
A+
Speaking as a ‘layman’ with only a basic knowledge of physics, this answer makes the most sense, as concerns the OP’s question. 
The gluon is theoretically massless, but it’s very hard to measure. Experimentally we can’t even be sure it isn’t more massive than the electron.
I thought relativity only said that you can’t travel AT the speed of light since any object with mass would become infinitely massive. But there’s not prohibition on traveling faster AFAIK. For example tachyons always travel faster, from the moment they’re created (if they exist).
So for example, if you could find a way travel faster than light by jumping from subluminal to superluminal, I don’t think that violates relativity. You’d essentially be “tunneling” through that barrier - for lack of a better term - rather than passing through it.
I wouldn’t take the idea of tachyons very seriously. They are logically inconsistent if they are able to interact with non-tachyons in any way; there is no way for a particle to “become tachyonic” (ie cross the light barrier).
Faster than light, mass and time become imaginary numbers. In other contexts in physics, that’s the equation’s way of saying it’s impossible.
Or telling us that the equations aren’t modeling reality correctly.
From a practical stand point it’s a very easy way to talk about large distances. How would you describe 85 light years in miles?
It does not explain why we traditionally say “light” rather than “radio”, or “gamma rays”, or just “electromagnetic radiation in general”, all of which consist of photons, and travel at c. I take the OP to asking why we say “light” specifically, and I think I gave the actual answer. It is a contingent historical fact that, when Einstein came up with the idea of c as the universal speed limit, he was primarily thinking of actual visible light (although he was aware that it is the same type of phenomenon as other electromagnetic radiation), and it was actual visible light whose speed had recently been measured. If Michelson and Morely had measured the speed of gamma radiation and Einstein had imagined surfing on a gamma wave, we would probably tend to refer to c as the speed of gamma.
Radio and gamma rays are light; they’re just not visible light.
But for the reasons pointed out by the OP, I prefer to refer to the Fundamental Speed as “c” (or just as “1”), rather than using the term “speed of light”. It is, after all, possible that the photon has a mass, or is in some other way constrained such that it doesn’t actually travel at c (just so very close to it that we can’t tell the difference with our instruments). It wasn’t all that long ago that neutrinos were thought to be massless, too, but we now know that they have a mass, just an incredibly small one.
The AC power you get out of the electrical outlet is modeled in physics using imaginary numbers. Nothing impossible or incorrect about my toaster’s functioning.
Not as English is spoken by regular people. (Not even as it is spoken by physicists in most contexts: not even, I very much suspect, as it is used in most physics laboratories and publications.) The OP’s question seems to me to be at lest as much concerned with colloquial English usage as with physical fundamentals.
In any case, would it not be just as true to say that light is gamma, just gamma that our eyes happen to be able to detect? The question, however, is why do we normally say “light” when referring to the universal speed limit (or why light should, in general, be semantically privileged over other forms of electromagnetic radiation).
Do physicists actually refer to it as “light speed”? Don’t they just call is “c”?
Surely the colloquial use of the word “light” is because that’s what people are most likely to be familiar with?
The word “light” is about as close to a colloquial synonym for “electromagnetic radiation” as you are going to get. Yes, in English the term often depends on context. If you say “let’s get some light in here” you are surely referring to visible light specifically. But if you say “speed of light”, you should understand it to be referring to “electromagnetic radiation”. Physicists don’t like to have to say “electromagnetic radiation” all the time (and it is something they need to say a LOT), and have found it natural to refer to it as “light”. For us, it is obvious from context, or else we say “visible light” if it is not obvious and we specifically want to refer to the visible spectrum. We very rarely use “light” alone to refer to the visible spectrum specifically unless it is particularly obvious from context. Now, the phrase “speed of light” comes from physicists. Physicists mean “speed of electromagnetic radiation”, not “speed of visible light.”
Whether or not that is true (and frankly, I remain skeptical that a physicist studying, say, radioactive decay, would habitually refer to a nucleus as emitting light rather than gamma radiation) the question remains as to why “light” rather that one of the several other words referring to particular parts of the electromagnetic spectrum is preferentially used, in popular accounts of relativity and the “speed limit of the universe” (I expect it is true, as Candyman74 says, that physicists these days mostly say “c”). The answer lies in the history of the matter, not in the physical facts per se.
I know that some physicists find it hard to accept that not all questions, not even all questions concerning the language and practice of physics, are to be answered entirely in terms of physical theory, but it is, in fact, the case.