Named after Third Avenue, one of Manhattan's avenues of 1811.
Third Avenue was commissioned in March 1811, as one of the numbered avenues of Manhattan's street plan. It starts at Astor Place and St. Mark's Place and crosses the Harlem River to Fordham in the Bronx. When the elevated railway above it was being torn down, there was a movement to rename the Manhattan part 'the Bouwerie', although, as Wikipedia notes, it had never been part of the Bowery.