Why do the elliptic curves recommended by NIST use 521 bits rather than 512?
I very much suspect it's to related to the fact that $2^{521}-1$ is prime. The previous similar prime is $2^{127}-1$ and the next such is $2^{607}-1$ so they're quite rare. Elliptic curve operations on such a field can be implemented somewhat faster than over another prime field with similar size but without this special form.

I very much doubt that serious thought went into this decision at all. If there are no breakthroughs in elliptic curve logarithm finding then 521-bit coordinate components are complete overkill. If there were a suitable breakthrough then there's no guarantee that 521 bits would be enough. 
Elliptic curves tend to be used when the size becomes a factor in the design of a system. It's hard to imagine what sort of constraints would have to operate to make a 521 bit curve order make sense.