Whoever gives him the better offer I guess...
Tywin was too deeply slighted by Aerys to ever join, the only thing I can think of is if Rheagar offers to Wed Cersei as a third wife...but I'm pretty sure Tywin would shoot that down from orbit. He might just back his son, out of a sense of family duty and settle for seeing a granddaughter on the throne.
Or sit it out entirely.
Presume that something occurs to make him less impressive for the White Cloak and Tywin perhaps arranges a marriage between him and Lysa after he had a talk with Hoster
How do things spiral from there? Does Tywin work with STAB(Stark, Tully, Arryn & Baratheon) and turn it into BLAST(Baratheon, Lannister, Arryn, Stark & Tully)?
Does Jon Arryn have a harder time joining up without a marriage which is a prerequisite to officialise an alliance?
Does the Reach and Dorne and the Crownlands decide "This is suicide, we're gonna fucking die"?
I doubt they'd declare for the Targs, Aerys was clearly bugfuck insane and the North had two modes of dealing with civil wars down South of the border.
"Ain't our problem"
And
"That's a real nice civil war you have there, be a shame if I ruined it by fucking killing everyone"
Cregan Stark's concept of being on the side of the Blacks amounted to "But I'm still going to ruinate everyone for the crime of making me ride my ass down to this shithole to deal with you fools".
Ned Stark is the
only exception to that rule besides his son and in both cases it was due to them being far more culturally Arryn than Stark.
Assuming Rickard and Bran aren't the "Hey, lets go absolutely ruin everyone's shit for annoying us with their civil war" variety. They'll probably just sit the entire damn war out.
Could see Eddard and Benjen riding South due to their emotional ties to the Vale but I really doubt they'll get involved.
Eddard might drag some young, up and coming nobles with him. The Great John, some Manderlys and maybe even a Roose Bolton whose looking to advance himself by shenanigans. There might be a "Company of the Wolf and Giant" or someshit..fighting as Bobby B's honorguard and given Eddard's skills as a general, I could see him becoming very well liked and well known.
I would be, fucking awesome to a brawl between Prime Great Jon and Lewin or Selmy...But that's the extent of it.
If Rickard swings the other way..hohohohohoho....holy
fuck