The model category of simplicial set valued presheaves on some category C, with the projective model structure, has a universal property: It is the initial model category receiving a functor from C (namely Yoneda embedding followed by the discrete simplicial presheaf functor). That is, for any functor from C into a model category there is a Quillen adjunction from simplicial presheaves on C to that model category "making the triangle commute".

One can think about this in analogy to the Yoneda embedding which makes presheaves on C into the initial cocomplete category (every functor is a colim of representables). Likewise simplicial set valued presheaves can be seen as the the initial hococomplete category (every object is a hocolim of representables). Note that this, unlike the first paragraph, is a statement not about the model category but about the homotopy theory it represents, which is maybe closer to what you are wondering about. Simplicial set valued presheaves with the injective model structure, or cubical set valued presheaves, would have the same property, but not the stricter one from the first paragraph.

This is a result from Dugger's article "Universal homotopy theories", see [his homepage][1], and while you are there by all means take a look at his expository paper "Sheaves and Homotopy Theory".

Summing up: Whenever you want to "go homotopical" on some category C, a good first step is to embed it into simplicial presheaves, then localize the model structure according to what weak equivalences you want to introduce in C. This is exactly what happens in $A^1$-homotopy theory, for example. To illustrate the difference between the universal properties stated in the first and in the second paragraph: Morel/Voevodsky use the injective model structure to start with, then localize by the $A^1$-equivalences. This is fine, as the injective and the projective model structures are Quillen equivalent and thus represent the same homotopy theory, so they do actually start with the initial homotopy theory containing schemes.
An advantage of taking the projective model structure instead (which is also perfectly possible) would be that you get Quillen adjunctions induced easily. E.g. the "complex points" functor from schemes to topological spaces induces a Quillen adjunction from simplicial presheaves with the $A^1$-model structure to the model category of topological spaces which is interesting to study; passing to the homotopy categories it allows you to associate to an $A^1$-homotopy type a topological homotopy type. Some theorems from usual homotopy theory can be recovered from their $A^1$-analoga this way.




  [1]: http://pages.uoregon.edu/ddugger/