It's not Rambo.
It doesn't even begin with an "R," like members of the first family (Renegade, Renaissance, Radiance, Rosebud.)
According to... well, I can't really say to whom, but Rahm Emanuel's code name is "Black Hawk."
The Secret Service allows its protectees to choose the code name used by its agents, but Rahm's was assigned to him during the Democratic National Convention, and he decided not to change it.
All other code names are assigned by the White House Military Office (through the White House Communications Agency.