Anyone tied to a faction can be made queen, and Lydia is heading The East faction, so she's definitely queenable.
On another note: I would say objectively that the only truly qualified candidates to be queen are Lydia, due to her extensive training along with powerful connections, and the well-read, willing to learn, and full of initiative: Valentina. Diana also has a quality resume, but she has rescinded any consideration, and with the rumors of her late husband, no intelligent king would want to risk her. As the heir to the Danish throne, although thereby trained for rule, Estrid is also a bad choice due to the issues of succession in both kingdoms. I know that was technically her plan with Marius, but that doesn't make it a good plan. Aelinia initially showed promise, displaying maturity and desire for such a role, but she now seems to ambitious (like possible future consultation with Diana about husbands), and her increasing drinking is starting to remind me of Queen Cersei.
As for the factions, since Gracchi has gotten both The West and The East to agree to work with them, I wonder if you can win all three factions by choosing a Gracchi queen? I personally doubt that ITRoy would allow this "cheat", so I imagine choosing Gracchi will either lose both, or you'll have to make a council appointment from one of the factions to win the two out of three. Objectively, since it seems pretty certain that MC will have to personally command the campaign against Queen Avila, it seems that the best faction to choose is The East with their Gracchi alliance, thus ensuring both a quick and decisive campaign and mutual loyalties.