big list - What are examples of good toy models in mathematics?

Perhaps this doesn't count as a toy model, rather a toy example. A nice basic example for GIT is n points in CP¹ under the action of SL(2,C). A lot of the usual elements of the theory look nice in this picture. For example, the Hilbert Mumford criterion shows that a collection of n points is semi-stable iff all points have multiplicity less or equal to n/2 whilst a collection of n points is stable iff all points have multiplicity strictly less than n/2.

It's also a nice example of the equivalence between symplectic reduction and the GIT quotient. If you fix a Fubini-Study metric on CP¹ and look at the action of the corresponding SU(2) you can ask for a moment map. Thinking of CP¹ as a coadjoint orbit in su(2)^*, the moment map takes n points to their centre of mass. Now the equivalence of symplectic and GIT quotients says that, provided we don't have two points each of multiplicity n/2, you can move n points in CP¹ by an element of SL(2,C) so that their centre of mass is zero if and only if all multiplicities are strictly less than n/2. (The case when two points each have multiplicity n/2 is special because of the additional C^* stabiliser.) In one direction this is obvious, but in the other I think this is a neat non-trivial statement (at least when n is large).