There are constants, it's just a matter of whether you are willing to accept them. Yes the speed of light is relative, thats been theorized for about 80 years. But there are other constants that are still recognized and no one has managed to come up with anything to disprove them, the that quickly come to mind are the Planck Constant and Hubble's Constant. For the purpose of not boring the forum readers who hate physics, I won't explain what those are, but those and other constants are recognized as existing. Oh and btw, the whole speed of light is relative thing, well thats really an insignifigant discovery in and of itself. It's the implications of that finding that are important, such as the fact that it leads to the relativity of time.
But anyway, thats irrelevant. The word constant and the appreviation const refer to without change. The word variable, and the appreviation var refer to having variation, or with change. They are exact opposites. For the sake of correctness, you should not call something a constant if it is not, and you shouldn't call something a variable if it is not. If you want to call %const a constant, than I can just as easily call 2 a variable. And if you are allowed to do such things, confusion will result.
Oh and also just on an abstract note, the speed of light is relative (according to Einstein) to a larger body of reference which is itself constant. Thus to make the observation that the speed of light is relative, you must accept that a constant body of reference exists.