As it is now: influence uses a system similar to sid meier's civilization 3 and nearly every 4x made after that.  You earn culture points, your borders expand, and if you optimize cultural production you can eventually swallow up other empires.  Kind of satisfying but also passive.

My idea: give players direct control over culture spread and culture flipping.  Ideally you would be able to do everything you can do with fleets, and whether you choose a shooting war or cultural conquest would depend on your civ's bonuses and who you're up agains

How I think it could work:

Culture would no longer passively grow your borders-instead you would have default borders defined by your controlled planets and starbases.

Influence points would be a stored resource that could be spent on various things.  They could even replace control and culture points.  One of the main things players could spend influence on would be to "culture bomb" hexes along their borders, allowing them to expand in specific directions instead of just slowly radiating outward.  Cost would depend on what's in the area you are trying to culture bomb

Flipping planets/colonies would work similar to how invasions work currently, except that instead of parking a ship in orbit for x turns, you would flip the planet by spending y influence points for x turns.  Planets would have to be within your borders and the cost/time it takes to flip a planet could be based on any or all of the following:

Treaties you have with that empire (open borders etc)

Approval

governor loyalty

Population in general as well as the number of citizens of your species on that world

Trade routes

Shared ideologies

Crime (if you have the crimelord trait)

and so on

with the other player being able to defend against culture flips

Closing borders

surveillance/secrecy/military buildings

secrecy/tradition perks?

culture starbases (which would now be used specifically to defend planets/hexes in the starbases radius, not to grow borders)

So ideally culture flipping would become a game of exploiting vulnerabilities in the opposing civilization, not the waiting game you have in most 4x's.

Influence points could also be spent on executive orders, cultural perks (you would still need awareness to unlock them, and if it takes a certain number of turns to unlock a perk you could keep players from buying out the whole tree in one go.)  This would reduce the different kinds of "paper mana" you have to keep track of and give you more flexibility about how to use stockpiled influence.  Influence could also be used for:

Tech theft (either stealing full techs or progress towards any tech known to the target)

Building starbases in somone else's borders

Removing their starbases/fleets from your borders?

Boosting individual trade routes (you get more, they get less)

Potentially causing their ships to defect

Im not sure how diplomatic capital works but you might be able to combine that with influence too.

And best of all: forcing another empire to join your war (you would have to already be at war with the target to prevent exploits and they would have to already have a bad relationship with said empire and not be in any other wars.)  this would allow you to do some stuff that happens IRL but that ive never seen in any game besides crusader kings.