The Mets would never allow it, even before the Yankees very likely did the same. They were created to fill the void left when the Brooklyn Dodgers and the New York Giants left for California.
I'd love to see the Dodgers back in Brooklyn, but what would a new team call itself? Also, where in Brooklyn would they play?
I don't see the likelihood of any teams relocating here, nor of the Mets moving to Brooklyn, so I don't see that happening.
The reason I write that it's unlikely that the Mets would move, is that Queens is generally neutral territory, since the Dodgers played in Brooklyn, and the Giants played in Harlem. If the Mets were to move to Brooklyn (fat chance with their new stadium, Citifield, being erected in 2009), that would mean that the team would officially be favoring the Dodgers over the Giants. Unofficially, the Wilpons do seem very much in love with the Dodgers and Ebbets Field.
I would name them from the past, the Brooklyn "Robins" or, more provocative and no less fun, The Brooklyn "Dem Bums". Brooklyn Bums.