Of course, classically, a subdivision of a simplicial complex K is  defined to be a simplicial complex L such that each simplex of L is contained in a simplex of K and each simplex of K is the union of finitely many simplices of L, and we can ask for a functorial version.  I don't know of a published analog for simplicial sets. It might be something like a functor Sd on the category of simplicial sets (maybe required to be induced from a functor from the simplicial category $\Delta$ to itself) together with a natural homeomorphism  |Sd X| \to |X|$.   Certainly there are known examples.  Segal's paper "Configuration spaces and iterated loop spaces'' introduces edgewise subdivision, and the first section of the paper "The cyclotomic trace and algebraic K-theory of spaces'' by B\"okstedt, Hsiang, and Madsen defines and exploits a variant of Segal's construction.  Tim, I leave it to you to see whether or not that suits your needs.