I won't address the second part of your question, since I'm not sure it's well-defined. But the first (as I understand it, "where does barycentric subdivision come from?") is a good question.
If you've read Hatcher's proof of the excision theorem, you'll remember he defines, for an open cover $\mathcal{U}$ of $X$, the chain complex $C^\mathcal{U}(X)$ to be the subcomplex of $C(X)$ given by singular simplices whose images are contained in an element of $\mathcal{U}$. He shows that the inclusion $C^\mathcal{U}(X)\to C(X)$ is a homotopy equivalence using barycentric subdivision---and excision, in the second form he states it, is obvious enough for the homology of $C^\mathcal{U}(X)$.
So there are two questions:
(1) What is the motivation for introducing the complex $C^\mathcal{U}(X)$?
(2) Why do we want barycentric subdivision to prove the homotopy equivalence?
Question (2) is easy enough -- we don't actually need barycentric subdivision, we just need something like it. We want to be able to send an arbitrary simplices $\sigma$ to sums of simplices contained within elements of $\mathcal{U}$, such that the sum in question is homologous to $\sigma$ (i.e. the boundaries cancel out). The obvious thing to do is to apply the Lebesgue number lemma, so we need some way of making simplices smaller by some definite factor. Furthermore, we need the map in question to be a chain map (to commute with the boundary map), which means it has to be built up inductively -- $\partial S=S\partial$ means that restricting the subdivision of an $n+1$-simlex to its faces must give the same subdivision as simply subdividing the faces. Barycentric subdivision is an obvious way to do this. You can motivate it yourself by trying to come up with a subdivision satisfying these two criteria for $1$-simplices and $2$-simplices; I bet you'll come up with barycentric subdivision. But it's by no means necessary -- there are any number of similar subdivisions. (You might for example define singular homology via cubes; then there's an extremely obvious subdivision!)
Question (1) is much deeper. Of course, excision in the second form Hatcher gives it suggests an approach like this -- but the intuition is a bit more exciting than that. What this approach means is that homology can be computed "locally"! Introducing singular cohomology, this idea leads directly to sheaf cohomology -- motivationally, if not historically.