Before anything else, I'd really like to have the AI improved in small but important ways:
- make the AI better at planning paths . At the moment it happily runs directly into most heavily guarded space. This is probably linked to the AI being bad a using sensors too. Instead it could stage some 'surprised' attacks.
- the AI is not good at predicting combat outcome either and goes into suicide attacks too easily: in one game it attacked a starbase it had no chance of ever destroying while the same (respectable) fleet could have been put to better use. Does the AI only looks at HP + attack + defence when evaluating military forces?? If yes that's very crude, and would explain the many stupid decisions I've seen. Computing odds is something machines are much better at than humans, so what not use it?

- the AI isn't goot at protecting mining starbase, why these usually provide huge bonusses (almost +40% when completely built out).
New cool tech: I really like the wormhole idea proposed earlier. This could be throttled to avoid completely chaotic games. The catch is that it only makes sense to have new techs if the AI use them reasonably well, otherwise it's a bit boring (e.g. Terror Stars).
Link hapiness and influence: this would make the game richer without introducing any new mechanism. The lower the hapiness, the easier it should be to flip a planet (since it's so bad here, why not try another ruler?), and the lower the influence, the lower the happiness should be (resident would be unhappy not to belong to another empire...).
Bug to correct: AI designed ships use greater capacity that current space (reported for instance here https://forums.galciv2.com/306683, looks like the designer does not take into account that components takes more space on bigger hulls).
GUI: better navigation in the ship mini list. Maybe using a tree like in SoaSE? (With different ways of grouping the nodes, class, expericne, sector, etc.) Could be expanded to the planet mini list (handy for very big games).
From the previous posts I quite like the 'spiral galaxy' option for galaxy layout.
Cloaking sounds good too (and easy to design, just like a negative sensor, reducing other ship's sensing range, e.g. is my cloaking is 2 and an enemy ship's sensor range is 4 I need to be within 2 square of the AI ship to be detected). Having the AI use it properly will be another matter!
A less radical alternative to the 'dwindling planet' suggestion could be to make asteroid and resource mining depletable, i.e. an asteroid field could disappear after being mined for too long. Could force some painful adjustments, and be fun for races at the height of their might.

Otherwise great game! I've never enjoyed a game so much since Civ 1.