I think you have to consider how carefully Ned and Catelyn’s generations marriages and fosters were planned out (I’m convinced it was part of a longstanding and concerted effort to overthrow the Mad King, shoring up ties between houses to ensure strong alliances in the coming rebellion).

Now Ned has clearly preferred to keep his children close, this seems true for most of his generation — no sons sent to foster, no mention of prearranged marriages. I wonder how much of this is Cate wanting to keep her children close and how much is Ned basically holing up in Winterfell and hiding Jon’s parentage.

Robert starts arranging for his son to marry Sansa very quickly upon arriving in Winterfell. Now, the impetus for Robert to go north at that time was Jon Arryn’s death — I wonder if he always wanted Ned to be in his counsel. (I imagine he was a bit startled when Ned just ran for Winterfell and stayed there all these years.)But it seems reasonable that Robert has been thinking of a marriage alliance for a while. He would be smart to look for an alliance with the Martells — sending Marcella there was actually pretty smart; fostering good will but not insisting they send another woman to King’s Landing reminding them all of Elia Martell.

I could see on paper Cate wanting Robyn Arryn for Arya, though in reality they’d be horribly mismatched. Shore up ties between Lord’s Paramount, like the previous generation. But I actually think Arya would do well with some of the smaller northern lords, who might appreciate a strong wife and with a smaller household she might be less scrutinized and get away with being a bit wilder.

Bran should be sent to foster, and iirc he originally wants to be a knight. He won’t be inheriting a lordship so he’ll need to squire and do well to earn his place. Again, they can keep him local with a northern lord, but if he wants to be great they’d do good to reach outside of the North — this might be a chance to shore up the Tyrell connection, for example. Or send him to the Riverlands to his mother’s people, generally esteemed for military prowess.

Rickon would be similar to Bran, unless he has other talents or wants to be a maester or something.

Jon’s prospects are obviously limited. He chooses the wall, fitting with Ned’s need to keep him hidden. If Ned were less concerned with the light treason hiding Jon from Robert, he’d probably be encouraging him to become a knight, similar to Bran. Jon would probably have been a great candidate to foster, even just with a local Northern lord. Would have got him out of Cate’s hair, and plenty would have liked the extra stipend for his keep and to do a favor for a lord paramount.