I agree with adding a single point of firepower to the flagship if it returns. It was a little pointless in mid-game, and merely wound up circling my homeworld as a makeshift defender in my games. Decidedly not being an expression of national pride and military expertise, but a laughable night-light blinking in the sky and waiting to die. Would also like anomaly events that add additional weapons, defenses, etc, and a fleet-boosting statistic added in general.
I think we should be able to designate a ship our "Flagship" at will, and gaining a mild boost centered around morale to the ships in fleet with her, but risking a global morale loss if we lose them. Kinda like real flagships- powerful, venerated ships that losing has very tangible consequences. Example: H.M.A.S Hood, WW II. But simply being told the starting bolt-bucket is your one-and-only flagship is a bit pale in my opinion.
I would like to see a system in game to really put that ship to use. The ideas above are great, thanks, but why not add things like contests on the different stats of the ships? Add races, naval duels and other things for the flagship to do, and give another reason for the players to strap guns on them when they can. Add missions and objectives for them, including escorts and dispute resolution, but also diplomatic fly-the-flag operations to help spread influence to your neighbors, and festivals and military displays to add experience to your ships and tourism money to your coffers. Special, risky black-bag missions could also be something cool, have great rewards, but huge diplomacy issues tied into them. That way, what you decide to do with your ships could affect your racial ethos as well as the random events.
I don't know, but I think there should be a tweak done to the flagship. There's plenty it could do, but more often than not it just sits about once the few anomalies have been snapped up.