He needs to learn to become a green seer. He starts being able to see the past via the weirwood (and possibly learns to communicate with afar via them). He also learns skinchanging.

It's been too long since I've read ADWD, I don't remember clearly what the hints were to where it might all be going. It's pretty clearly tied with the "cosmic conflict", though. Jon, Arya, Bran are all enmeshed with "gods".. Jon via Melisandre, Bran with the Old Gods and children of the Forest, and Arya with the God of Many Faces (the god of death).

It's highly suspected that Bran might learn the secret of Jon's real ancestry, among other things. He's already got a vision of Ned praying to the tree for what seems to be Catelyn's forgiveness and his wish that the boys grow up to be as close as brothers (He seems to mean newborn Robb and his fake-brother, probably cousin, Jon). Then Bran has at least another vision that fans theorize was showing a young Lyanna and Benjen (Bran compares her to Arya, who is said to look like her aunt). So it looks like GRRM is setting up for Bran to learn the truth soon, and I guess eventually pass it on to Jon. He's also the one who in time can reveal a lot more to the reader about the true nature of the big conflict, as he can see the past.