Let $X$ be the sort of topological space for which it makes sense to talk about the [intersection homology](http://en.wikipedia.org/wiki/Intersection_homology). Fix a perversity $p$, or just take $p= 1/2$ if you like. 



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Is intersection homology the usual homology of something else?
I'm not sure this is the kind of answer you want, but if $X$ has a *small resolution* $f:X' \rightarrow X$ (so $X'$ is a manifold and the dimension of fibers is sufficiently small), then there is an induced isomorphism $IH\_{\ast}(X) = IH\_{\ast}(X')$, and because $X'$ is smooth, the latter group is $H\_{\ast}(X')$.


More precisely, a proper (birational) map is *small* if the set $${x \in X | \dim f^{-1}(x) \geq r }$$ has codimension more than $2r$; such maps induce isomorphisms on IH.


Of course, there is nothing natural about $X'$, nor do small resolutions necessarily exist (see the comment below by Mike Skirvin for an easy example). Hopefully someone more knowledgeable about IH will have something to say.


