Alright, I'm relatively new to this game, but have been playing strategy games for a while and have a few things I'd be interested in seeing.
Hotseat multiplayer. I honestly don't care about LAN or Internet multiplayer that much.
More hull types and sizes, perhaps a hull type for each civilization. Also, someone suggested a chassis-editor for the hull types, and I like that idea as well.
The anomaly system is cool but on standard and smaller maps most of the anomalies are gone by the time you're able to build new survey ships or put the survey module on ships. If anomalies or other events gradually got reintroduced into the game?
In the opening video it says something about the civilizations using stargates to move ships from one place to another, functioning almost identically to the Star Gates from the movie of the same name. Yet we don't see them in the game at all, and it'd be a great way to get around quickly on a Gigantic map in the mid/endgame.
An expanded tech tree. More government options, more ethic-based options, perhaps a general defense system for ships.
As some people have suggested, ways to create large-scale space stations, terraform planets directly (ie transform unusable tiles into usable tiles), or create planets (obviously a lengthy and involved task).
A way to get some use out of class 0 planets, either by the above method or by allowing military bases, etc. to be built on them.
A bonus for having a planet with a moon that might have resources, etc. (the moons are there in the game but serve no purpose other than to look pretty)
Characters or special forces, that can station on a planet, gain experience like ships, and provide special bonuses or advantages.
A more involved planetary invasion system, and a tactical battle system for fleet battles.
Pirates, aliens, and minor nations serving a greater role.
A more involved Ethical system. More choices to make, penalties for changing ethics too quickly or too much, bonuses for staying consistent in alignment for long periods of time.
More distinction between "fighters" and "capital ships"; it's mentioned in the manual but in play fighters are just smaller capital ships. Perhaps carrier-style ships, that can carry fighters or groups of them? Or even better, a way for fighters to have their own logistic system, called "starfighter points" or something, and for larger ships to have a "fighter bay module" that grants them a certain "starfighter point capacity, so fighters don't count against a fleet's capacity but you have to customize or specialize ships to carry them.
A few things inspired directly from other games that I'd like to see here:
A "Civilopedia"-like reference section where I can find information about the ships, technologies, planets, etc.
Also from Civilization, a more indepth diplomacy, trade, and espionage system.
From Star Wars: Rebellion - the ability to perform "missions" on planets without having to invade or bombard. For example, you could perform subversion missions to reduce enemies' approval ratings, sabotage missions to take out key starports or industrial sectors, etc.
Also from SW:Reb - More random events on planets, like archaeological discoveries (providing research bonuses), disasters (reducing available tiles and/or destroying improvements), or resource finds (providing manufacturing, food, or economic bonuses).
From Spore (I know it's not out yet, but still) - A way to upload my own civs, ships, campaigns, and mods and download others' civs, ships, mods, and campaigns from the game itself.
As I said I'm new to this game but it's a great game and I hope to see a great expansion for it.