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.

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 $\operatorname{Sd}$ 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 $\lvert\operatorname{Sd} X\rvert \to \lvert X\rvert$. 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 Bökstedt, 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.