What is the word for how polyprotic an acid is?
The term that I learned was (ironically) "basicity."

I'm not sure if that's the standard term, or even if there is a standard term, but this refers to how many bases you need to neutralize the acid completely.

EDIT: Thanks for the correction from LDC3: the "basicity" of an acid refers to how many bases you've already added to it. Thus, $\ce{NaH2PO4}$ would be "sodium phosphate, monobasic," $\ce{Na2HPO4}$ would be "sodium phosphate, dibasic," and so on.

Again, not sure if this is standard terminology.