Out Of My Vulcan Mind wrote:
Even if legitimized, Jon would have to turn his back on his vow to the Watch to take the throne, though, so that's another complicating factor.
Robb made someone his heir and put it in writing, but it wasn't revealed who he ended up naming.
IIRC, Stannis offered Jon Winterfell, so I guess there's a way to turn back on the vow and not being beheaded for it. Anyway, bastards can be legitimized by kings - the example was Bolton's bastard who was legitimized following the Red Wedding.
I don't think there were many possibilities for Robb - he thought his younger brothers are dead and he has no idea what's going on with his sisters. His father is dead, his uncle is missing beyond the Wall and he basically has no other possible candidates besides Jon. As far as he knew - he and Jon were the only male Starks left alive.