It was named after an individual named Jonas Bronck, but some maps and documents from before it was incorporated into the city refer to it in relation to the river also named after Bronck, so the Annexed District of the Bronx and Borough of the Bronx. Officially, it's just called Bronx so that's how the Postal Service and Bronx County refer to it.

http://en.wikipedia.org/wiki/The_Bronx#Etymology

Boom