If like me your a story driven gamer (which I think most GalCiv gamers will be to a certain extent), then every sandbox game is the chance to immerse yourself in a unique history and Galacto-Political adventure. I've had a few ideas to try and enhance this as the game already does this very well in and of itself.
1 - Little bit of plagiarism here as I think this was mentioned in one of the Dev Streams - have the ability to name areas of the Galaxy. Would be really cool if you could select a group of tiles (be it large or small), name it, and then have the map highlight that area whenever you mouse over it. And if the AI could reference your area names (in diplomacy for eg) that would be even better.
2 - Add the ability to name periods of time - this would mainly be for wars but you could use it for anything you wanted, be it periods of increased trade (The Golden Age), or Research (The Enlightenment). And again, the AI could reference the names you have given to present/past conflicts.
3 - When any treaty is agreed (open borders, trade, alliance) have the game randomly pick a planet (yours or theirs) that it was signed on. This could give the planet some kind of influence bonus, but in terms of immersion I'm thinking of giving the player the ability to name the treaty ie. The Khitomer Accords, or a more terrestrial example The Geneva Convention etc.
4 - Using all of he above you could then make 'The News' a lot more detailed and specific. It would be good if you got notification of made or broken alliances this way, plus declarations of war, war news (who's winning/losing), systems taken etc. All using specific location, time, treaty and war names. To be clear, this would apply to all factions you were aware of, not just in relation to your faction.
5 - This is just a cosmetic thing - personally I don't think 1 week per turn is very realistic and therefore breaks immersion. This is only personal preference but I think 2 or 3 weeks per turn feels right to me. Of course this could just be me, and I'm sure others would have different preferences - so how about another game setting where you can change how many weeks pass per turn? For clarity, this would not change the game mechanics in any way - purely for appearances.