TheRegarding finite field versions of quantum mechanics, the following paper is a good place to start:
"Modal quantum theory", by Schumacher & Westmoreland, arXiv:1010.2929
In this paper, the authors present a discrete model of quantum theory that is similar to standard quantum mechanics, but it is based on finite field-valued amplitudes instead of complex amplitudes. The theory is surprisingly rich: it allows for entangled states and contains versions of the no-cloning theorem and Bell's theorem. It also has a distinctly different flavor when it comes to probabilities, namely (as the title suggets) it deals only with modality, the possibility or necessity of an outcome, rather than probability measures.
There is at least one other program for studying toy theories which comes to mind. These go by the name of "general probabilistic theories" or "convex operational theories". Here the idea is to replace quantum mechanics with a general convex space which becomes the space of quantum states. Then you decorate this space with things like tensor products to define composite systems and so on. It lets you ask questions about why quantum mechanics is special, since only some of the things we think of as being quintessentially "quantum" can be done in these other theories. In this regard, they do indeed shed important light on the nature of quantum mechanics.
The names I most associate to this approach are Barnum, Barrett, Leifer and Wilce, though I may have forgotten some others. Searching the quant-ph arxiv will turn up some papers for you. One which I've read personally, and it should get you started, is
"Information processing in convex operational theories", Barnum & Wilce, arXiv:0908.2352