More brainstorming:
- Request revocation of titles owned by characters they hate (not sure how you'd determine this, but eh)
- Changing investiture law (imagining tug of war requests between faction and Pope)
- Demand liege gives up some of their demesne (if it's particularly large, possibly?)
- Demand liege divorce current wife in favor of someone else? (I don't know, could get really complicated)
- Offer to change religion or culture to match liege, or demand liege changes either (this seems unlikely to be a good idea)