1. Fix the broken improvements... yep, that's another 2c for the Mind Control Center.

I would also include the planetary defense improvements - some of them claim to give a bonus but don't actually show the little "defense +x%" icon in planet menus, and defense bonuses don't show up on the details screen for that planet. I can't tell from advantage numbers whether these improvements actually work.
2. Espionage overhaul (part I) - I understand this is already in the works in some form. My first gripe is that sabotage doesn't scale well. Most systems in the game have linear cost with exponential benefit - doubling your warship fleet costs twice as much and uses twice as much in maintenance, but is more than twice as good, since even with equal logistics there is an advantage in having numbers. Likewise with influence (high influence makes planets flip, yielding more influence) and infrastructure (a strong economy can afford to build improvements and starbases that make it even stronger). On the other hand, spy sabotage has exponential costs and linear benefits. In order to have twice as many planted spies as an opponent, I must spend much more than twice as much in time and money. In order to do the same against multiple opponents, I need a hugely larger investment.
3. Espionage overhaul (part II) - The second gripe, and something that might get overlooked, is that raising your espionage level gets you a lot of information, but it's mostly useless and almost all terribly disorganized.
For example, I can read that "our spies have determined that the Drengin spaceship so-and-so has a defense rating of 3 and is headed for Someplace-I've-Never-Heard-Of", or something like that. This line in the report is never of any conceivable use to the player - for one thing, if a particular ship or fleet is close enough to player holdings/units that its nature and destination actually matter, by definition it is already within sensor range.
I would really like it if the espionage report gave me easy-to-understand and, above all, actionable intelligence. It would be great if espionage could tell me, say, what the computer player's current plan for victory is. For example, I just played a game where the bordering Drengin had much better weaponry but were too busy with other civs to bother little ol' me... for most of the game. Then they decided it was time, demanded tribute, and attacked when I refused. I would really love it if espionage could tell me in that situation something like, "The Drengin plan to conquer the galaxy through force, and are currently (struggling/succeeding/plotting their next move/etc.) We are viewed as (a potential threat/a potential ally/fresh meat)." When the AI changes its mind about its plans, perhaps a message could come up, "Our spies have determined that the Drengin plan to conquer us next. We (can easily best them/need help/are doomed). We should probably (plan our next vacation on Drengi/search for allies/try to buy them off)."
Likewise it would be nice to know the relative strengths and weaknesses of foreign planets - i.e. "Westerfield II is the center of the Drengin economy. Xassica IV is the home of their Manufacturing Capital." etc. You can get some of this data now by looking at foreign planets when you place spies, but you have to actually page through each and every planet in order to get the data - I would like to have an easy-to-read summary.
4. Easier starbase construction. It would be nice if I could work with starbases like I work with planets, where you see a clear list of what is available and can place desired items in a queue. I can never remember which modules I already have on a starbase and which I want, and it's super annoying to have to send a zillion constructors gallivanting around the galaxy to get everything done. I would like to have something like how trade works. In the same way that you build a freighter, click on an alien world, and then everything just works from there on out, I wish that I could build a constructor, pick a point for a starbase, choose the desired modules, and then
never ever click on another menu ever again and have everything work out automatically.
5. Please teach the AI that a ship with defenses but no weapons is not a good defender. I suspect that the AI takes the settings last used by the player in the shipyard focus for weapon/defense balances and just keeps them irrespective of what war technologies they have actually researched; that seems to be the simplest explanation for the awe-inspiringly dumb ship design decisions I've seen in my latest game.
Two civilizations tried to wage war with me even though they had exactly 0 ships with actual weapons; they would keep building defenders with shields but no weapons, and I just kept blowing them away. I had a ship that was the first fighting ship I ever made and equipped with the worst weapons in the game. That one ship just kept flying between clustered planets, where I would sent it to blow up the unarmed defender of planet A, then the unarmed defender of planet B, then back to A for the newly-launched yet still unarmed defender etc. until I finally came in and crushed them all with troop transports and sent my ship to the next planet cluster.
After something like 3 game-years of continuous badassery, that one small-size ship had gained so many experience levels that it had 45 hitpoints. And during that entire time, in which I conquered something like 30 planets, neither opponent civ figured out that it might be a good idea to build ships that can shoot back. To make this whole experience all the more insulting, through most of this both civs were convinced that they were totally winning the war - even though they had never won a battle and all their citizens had developed cancer from the falling fragments of their "warships" that I had blasted into radioactive smithereens over and over and over.
6 [Bonus Round]. Speaking of which, it would be nice if citizens actually noticed when a planet's rivers start to run red with the blood of their fallen comrades (or green or purple, as the case may be). If you invade a 16-billion-population planet with 40% approval and wipe out most of them, the survivors will suddenly all become ecstaticly happy at the extra room on the planet and approval shoots up to 100%. I guess they don't mind the smell from the
12 billion decaying corpses of everyone they have ever known or loved littering the planet. Repeated military losses in orbit or on the ground should probably damage morale, perhaps with the effect dampened by civ loyalty.