An update to yesterday's ramble: someone suggested checking out the Post Office in my zip code (11216) to see what they call the area. This only complicated the issue--the local Post Office is "Brevoort Station." By zip code, this means I live in Brevoort, NY. The Brevoorts were one of the original Dutch families of New Amsterdam (the Parks people have a nice bit on them here), and a nearby NYCHA project and park bear their name. Their connection to Brooklyn seems to be through the Lefferts-Brevoort Homestead. Another root of the Post Office title might be Willam Augustus Brevoort Coolidge, for whom Brevoort Place (just north of Atlantic) is named.
I haven't met a soul that says they live in Brevoort, though a local artist does use the name.