I've always felt that the flagship should be much more important than it is. It's nice that anomalies respawn now, which gives the flagship something to do for the entire game, but it seems like a ship that mirrors the Enterprise or another species representing ship should have some important things to do. What I'd love to see is:
--Missions! Flagship stops at planets and picks up missions (sort of like Puzzle Quest) where they can get a reward for transporting a diplomat to another planet, bringing scientists to witness an event at a certain parsec, delivering medical supplies to a planet/fleet/etc. The possibilites are endless here. Just use the plotlines from sci-fi TV shows for inspiration.
--Rewards for completing missions could be any number of things, not just cash. Discounts on construction, new technologies (some exclusively gained from missions even), influence points, changed relations between races, ascension points, etc. The possibilities for rewards are many as well.
--No cutscenes would be necessary, simply a new screen with text for each mission and a generic mission complete screen.
This seems like it would be easy enough to implement into GalCiv2 even, but if it proves impossible or too much work than I'd love to at least see it in GalCivIII.