Simplicial Model of Hopf Map?
There is a paper [MathSciNet] of Madahar and Arkaria called A minimal triangulation of the Hopf map and its application. They find a triangulation from a 12-vertex 3-sphere to a 4-vertex 2-sphere. The minimality is in Section 6.a. I hope this is useful.
Now, this gives the map the structure of a map of simplicial complexes. Choose an ordering of the vertices such that the map in the paper respects the order. This then gives you a model of the map on finite simplicial sets.
Here is one thing to try. Start with the smallest simplicial model for S1 (the 1-simplex modulo its boundary). Take the free group in each degree (but force the basepoint to be the identity). The resulting simplicial group FS1 is a model for ΩS2; furthermore, being a simplicial group, it's a Kan complex. Thus, we know there must be some map f: S2->FS1 which represents the generator of π2ΩS2; the group of FS1 in degree 2 is not too big, so it should not be hard to write this down explicitly (I haven't tried, though.)
Of course, you really want a map S3->X, where X models the 2-sphere. Since FS1 is a simplicial group, let X=BF1, it's classifying space. X is a model for the 2-sphere, and I expect that if you examine it closely, you will see the "suspension" of f corresponds to some explicit 3-simplex in X, which is your model.
I'm not sure this counts as a "combinatorial model", of course.
(I have a vague memory that Dan Kan did something like this in one of his papers in the 50s. Is that right?)
Here's a different answer. The Hopf fibration S3 -> S2 is a principal U(1)-bundle, which means it is the pullback of the universal U(1)-bundle along a map S2->BU(1).
There is a simplicial model E->B of the universal fibration over BU(1) which is a Kan fibration: since BU(1) is K(Z,2), you can take B to be a simplicial abelian group associated to the chain complex C concentrated in degree 2, and E is the simplicial abelian group associated to an acyclic complex A which has a surjective map to C. Now pull back along S2->B and get a bundle Y->S2, and there you are. The simplicial set Y will be a model for S3.