In a restart, I do think that we should use the original session to at least build back story. For example, I plan on beginning the game with an intense rivalry/dislike between the House of Rose and the House of Piast due to an ancient slight that neither House has forgotten (assuming Sterk is fine with doing that, of course), and I think the sniping between the House of Vermandois and the House of Hapsburg should not be set aside, either. Of course, since the disagreements between the German and the Frank are territorial in nature, they would likely manifest again anyways. Talking things out would also give us the opportunity to work in deeper relations between Italy/Aragon and those of us who played in the first session.
I'd also like to suggest that we revise our characters somewhat for this start. In particular, I think we should try to bring their stats in line with how we have been role-playing our characters. In my own case, I feel I should probably lower my dimplomacy stat, since I have learned that good King George is not a very diplomatic person. He's also more war-like and cruel than I initially imagined him, what with his love for persecuting and torturing the Welsh. I feel that Sterk would do well to revise his character and make sure he has the Proud trait, since most agree that he is playing a very proud person. Solmyr might consider both Lustful and Indulgent, since those were his most prominant traits during the RP aspects of the game. I'd even go so far as to suggest that Hyzhen go about trying to build his character as Modest and Wise from the way he played Gaspar. And so on.
I think there is a natural tendancy among us to try to min-max a bit when we build our characters, and I'd like to see us take advantage of a restart to make our characters more about the RP aspect of the game rather than the mechanics of the game.