When a tectonic plate subducts, does any of the subducted material melt, or is it just the mantle above the subducted slab that melts?
This is a very good question Inkenbrandt. It is quite common for even some graduate students to think that the majority of partial-melting comes from the slab: when it is the the mantle above the fore-arc, in the volcanic region, where most of the partial melting occurs.

Lets think about this mathematically: you say yourself that you realize that subduction hydrates the mantle, lowering the solidus and allowing for more partial melting, so what happens to the slab? Well it has to dehydrate, making it much harder to melt the slab. So to answer your question: Probably a very small amount in the upper mantle, but if you are a mantle plume believer, the partial melting of a slab provides a great mechanism to 1) displace hot mantle so it rises and 2) a source of meltable material.  The hydration of the mantle from subduction happens on a much much faster timescale than the melting of a subducted slab. In this paper, the authors provide a mechanism for the volcanism in Yellowstone that relies on two things: the Farallon plate being ripped apart by rapid mantle flow and the cut off piece of the subducted slab to melt. Geochemical data supports this, and it turns out this is a very important paper.

I hope this answers your question. In geodynamic models interested in examining the melting structure at subduction zones, we largely ignore slab melting unless we are specifically looking at it. For almost everything else, it seems to be pretty negligible.