A fun exercise...
I wouldn't want to try predict the outcome, given the complexity of the story, and Martin's willingness to blindside us, so what follows is not my prediction, but rather, say, my "wishlist" of the fate of the Starks;
Jon Snow
He has been killed, but Melisandre uses the well-established abilities of the Red God to revive him. He uses the technicality of his "death" to honourably free himself of his Nightswatch oath. While he unites with Melisandre for a time, his fate is ultimately tied to Daenerys, his aunt. Together they are responsible for throwing back the White Walkers, with the support of the Stark family and their supporters.
Catelyn Stark
As Lady Stoneheart, acts as a pitiless counsellor to Sansa Stark, and directs her towards an utterly mercilessly treatment of her enemies. After all the Lannister children have been killed, Catelyn will be the one who kills Cersei.
Sansa Stark
My personal favourite, and the most ripe with potential. As she grows and matures, her traumatic experiences and involvement with Littlefinger harden her into a ruthless, cunning, and clever player of "The Game". She outplays Littlefinger, becoming a dominant ruler (either a Queen of a divided Westeros, or a major noble), and deftly leads the reunited Stark siblings in a ruthless and bloody war of revenge on their enemies. She honours her marriage to Tyrion, and they form an unlikely but perfectly matched alliance. Tyrion somewhat counters Catelyn's influence on the "Ice Queen".
Arya Stark
Reunited eventually with her sister, Arya becomes a shadowy and emotionless killer, slowly working through her list, and acting as an extension of her sister's will.
Bran Stark
Merges with his tree and is elevated to a higher level of awareness, brings forth the Children of the Forest to ally with Jon and Daenerys in their battle against the White Walkers.
Rickon Stark
A wild and out-of-control monster, on occasion let loose by his family to terrorise their enemies, but eventually has to be caged and/or killed because he can't be controlled.