In first ed, there were four cards called "The Iron Throne". Each was different, but at the core level this card game is simulating controll over the Iron Throne of Westeros. Now, I played L5R back in the day, and they had a mechanic called "The Imperial Favor" which represented the control at court.
Using a similar idea, what if while you were deckbuilding you picked a version of the throne to use if you win dominance?
For example, if I win dom I stand my version of the Iron Throne and then can kneel it to use the effect at anytime. Other cards would not be able to affect it, sort of like a house card or agenda, unless they specifically mentioned the "throne" trait. If someone else wins dom, all other thrones in play become knelt if standing, and then the player that won stands their throne.
This makes the dominance phase important, but mabye most thrones let you kneel them to claim a power or draw a card, and then mabye another variable ability. Lany now can cancel abilities, a Targ one may let you initiate another power challenge, Stark may let you stand a defender, etc.
For however many cards they had in first ed, I think this could be an interesting idea. Some thrones could be neutral, but faction specific ones could be added later. It makes the Dominance Phase more interesting, and could affect deckbuilding similar to how agendas work.