In Einstein's paper, "On the Electrodynamics of Moving Bodies" light is used to synchronize the clocks of two different points in space. Thus if a frame S' is moving relative to frame S, an observer in frame S will observe that there is time dilation in frame S' because of the use of light to attempt to synchronize a clock in S with a clock in S'. However, it seems rather arbitrary to use light to synchronize clocks since it seems like you could devise a method of synchronizing clocks that involved the use of some other object whose speed is not constant, thus freeing you from the problems that come from using light (whose speed is constant) to synchronize a clock between the frames. What am I missing?
Indeed, the choice of light for synchronization is arbitrary and not necessary. Other conventions could have been chosen. This concept has been explored extensively in the literature, including especially Reichenbach.
The synchronization convention used by Einstein is widely regarded as the most natural and reasonable one, while recognizing that it is a convention. So what happens to our understanding of relativity if we dispense with this convention? Luckily, the mathematical framework of pseudo-Riemannian geometry allows us to consider such things. We can investigate arbitrary coordinate systems, including arbitrary synchronization conventions, using the concepts of coordinate charts and tensors.
It turns out that there are still many relativistic effects that are not dependent on the synchronization convention. A version of time dilation would survive, the relativistic Doppler effect, it would still require infinite energy to reach the speed of light, the twin paradox would still happen, and electromagnetism would still be a relativistic field theory.
The relativity of simultaneity would be different as would length contraction, not absent but more general. Of course, it is unclear from a historical perspective how relativity would have been developed without Einstein’s convention.
I would like to give a short summary of my views on the subject. Dale is right that
the choice of light for synchronization is arbitrary and not necessary
But there is a point seldom remarked. Einstein's synchronization (ES) is physically meaningful. I mean that it's meaningful that ES is possible.
Assume you have three clocks in different locations, A, B, C. Decide to use ES between A and B, and also between A and C. Then compare B and C: will you find them ES synchronized? Here there is no arbitrariness, no convention: either they are synchronized, or they aren't. Only experiment may decide. And the experiment speaks in the positive: ES is transitive.
Then I can't see any reason to adopt a different synchronization, which requires greater attention and a more complex mathematical apparatus. An analogy can help. It's well known that in most practical and scientific applications we are allowed to assume space obeys euclidean geometry. This means that cartesian orthogonal and isometric coordinates are possible. No doubt this is an arbitrary choice - we could use curvilinear coordinates as well. But at what advantage? Cartesian coordinates bring written in their structure - so to say - the Euclidean character of space, with the simplest possible formula for distance: $$ds^2 = dx^2 + dy^2 + dz^2.$$
It's true that we may take recourse to Riemannian geometry also in a Euclidean space - and thinking of spacetime, also if it is Minkowskian, as in RR - but to what avail?
Let me repeat. The only physically meaningful statement is that ES is possible (it's transitive). Given its advantages, I can't see any reason for other choices. And actually I never saw other synchronizations used, outside epistemological discussions. Even there, other synchronizations are always introduced as modifications of ES. Apparently there's no independent way to define one.