For a given ally, I don't always get the same character to borrow, but it's usually one of the same two or three, and often it's the same one. It seems like it could be a random selection from among their most powerful characters. I've also read that it might be the character the ally most recently used as their leader.
I've noticed that my most recent team affects what's available. For example, if I had Chewie on my last team (for that squad) then no Chewies are available to borrow. That makes sense, in that I can't have two Chewies on my team, but they could just remove him on the squad selection and force me to replace him with someone else. Anyway, that's at least one thing that affects who can be borrowed, but is there anything else we know for sure?