This question may be somewhat relevant: [Small simplicial complexes with torsion in their homology][1].  David Speyer's answer there shows that one can build a simplicial complex $X$ with $H_1(X)=\mathbb{Z}/p$ where the number of simplices of $X$ is $O(\log(p))$.  It seems unlikely that one can do much better than that in the world of simplicial sets. With CW complexes, you only need a single $0$-cell, a single $1$-cell and a single $2$-cell.  This gives an initial picture of how the simplicial set version of the question might deviate from the CW complex version.


  [1]: https://mathoverflow.net/users/297/david-e-speyer