A wholly different set of "named urban legends" (in order of time):
Allegedly, Jacobi came to show Gauss his cool results on elliptic functions. Gauss' response was to open a drawer, point at a sheaf of papers, and say: that's great you are doing this! I have actually discovered these results a while ago, but did not think they were good enough to publish... To which Jacobi responded: Funny, you have published a lot worse results.
When the logician Carnap was immigrating to the US, he had the usual consular interview, where one of the questions was (and still is, I think): "Would you favor the overthrow of the US government by violence, or force of arms?". He thought for a while, and responded: "I would have to say force of arms..."
Finally, on the graduate experience front, it was rumored at Princeton that Bill Thurston's qualifying exams at Berkeley were held as his wife was in labor with his first child -- the department refused to change the date for such a minor reason! I have just asked him about this, and it's true...
EDIT A certain (now well-known) mathematician was a postdoc at IHES in the late 1980s. Call him R. R comes to lunch, and finds himself across the table from Misha Gromov. Gromov, very charmingly, asks him what he was working on. R tells him, Gromov has some comments, they have a good conversation, lunch is over. The next day R finds himself across from Gromov again. Misha's first question is: so, what are you working on now?