Okay, this is not letting me go. I thought a bit and this is what came out of it:
Influence points - would basically replace culture for border expansion, which still happens automatically (mostly, more on that later). Further, they can be traded with other players, similar to gold (but only in lumps, not per turn). I could also imagine buying great persons/naturalists with it, like with faith, but I have no idea how that could impact balance.
Border expansion - happens automatically whenever a city reaches a treshold with locally produced influence, as it is with culture now. However, any influence produced in a city also goes to the global counter, even if the borders expand. There will be ways to directly buy tiles with influence though (more later

)
Envoys - are produced by the global influence (of a civ). Again, this doesnt use up the stored influence, it's tresholds again. The tresholds might change with more advanced government forms, as it is now. Governments could also produce global influence for a civ, without it being bound to a city (or it being bound to the capital, just a spontanous thought).
World Congress - depends on how it is implemented really, but I could imagine something like influence being used (this time used up for real) to back certain resolutions with more "power", making other civs (or city states?) sway in your direction. Or just additionally envoys for more city states? Well, however it is implemented.
I personally would prefer a system where city states vote for themselves, but you can influence them. For example, a religious city state would be more inclined to vote for their religion as world religion, spending influence points on them could convince them otherwise though. Naturally, as several players can spend influence points, it can look like this:
Example
World Religion: Hinduism (Founder: India) There are other religions, but in this example, Greece hasnt found one, but also doesnt follow Hinduism. Russia has founded another religion and China follows Hinduism.
India votes Yes - backs it with 400 Influence.
Greece votes No - backs it with 100 Influence.
Russia votes No - backs it with 300 Influence.
China votes Yes - doesnt back it.
Jerusalem doesnt follow Hinduism and thus would vote No. Let's say it's worth 200 Influence points to them. In other words, they back their own vote with 200 Influence. To sway them to Yes, one would have to overcome this 200 Influence, plus whatever all other players added to this. In this example, it would be the following:
100(Greece) + 300 (Russia) + 200(Base) = 600
So they are 600 points for No, while India only backs with 400 points, resulting in -200 for Yes. Jerusalem would stay No.
Kabul, on the other hand, also doesnt follow Hinduism, but they are militaristic, so it's only worth 50 points to them. Again:
100(Greece) + 300 (Russia) + 50(Base) = 450
With the 400 points invested, India would still not be able to get the Yes out of Kabul (-50). But wait! Kabul is allied with China, and they voted Yes! Naturally, a suzerain should get a small bonus, let's say it's worth 100 points. Like this, the Yes vote would go up to 50 and thus sway Kabul's vote.
So much for my little example. ^^"
But where do Influence points actually come from!
As already mentioned earlier, governments could provide some influence points, further, the palace could offer a nice starting amount for the capital. Apart from that, there would be buildings. I kinda don't think it should come from population, too.
Monument - as the early culture isnt needed for border expansion anymore, the monument could provide a small baseline amount for new cities (the culture loss being offset by a slight increase of culture/population).
Diplomatic District - Main infrastructure for producing influence. Baseline, it would provide, like, +1 Influence points and +1 for each adjacent district/city centre. (All numbers and names not final obviously

) Further, the district allows buying of tiles with Influence points in the city it is constructed in. This should be cheaper than buying them with gold btw. Also, delegations and embassies in other civs could yield influence.
For the buildings, as mentioned in my first post, there could be:
(Tribal) Meeting Hall - Next to offering more influence points, this building would allow either buying any tile (in range of a city and next to an already owned tile) in a 9 tile (or so, I feel like it should be greater than the usual 6) radius with Influence points, or just giving the ability to buy tiles with influence to any city in range. Personally, I think the former is more interesting.
Realm Assembly - Makes buying tiles in the radius cheaper maybe? I haven't had a striking idea here yet. Could also swap the bonus for this and Meeting Hall around.
Embassies - Could maybe allow buying tiles of city states within the radius. Much more expensive than buying regular tiles and cant buy tiles in a city's first ring. I'd also consider buying tiles of other civs, but I think that should be a mutual trade deal if anything, and thus would be very tedious to do for each tile on its own.
I think the District and buildings should have their own leaf civics to unlock. District and Meeting Hall in early classical era. Realm Assembly in late Medieval and Embassies in early Industrial era for example. Accompanied by a fitting set of social policies (accustomed to the diplomatic and/or economy slot).
So, what do you think of my, ahem, little brainstorming? ^^"
Edit: Okay, I just had another idea for the Realm Assembly. Let's maybe give it the bonus from the Meeting Hall for now, to buy tiles with influence, and give the Meeting Hall the bonus of increased influence generation (or, cheaper automatic tiles) for cities in range, or on the continent, or in cities smaller than the city with the Diplomatic district. The latter is my favourite btw, as it kinda makes it a capital tool, at least for the early game.
Edit2: Actually, no. The whole reason I gave the Meeting Hall the tile buying ability was to have an early use for Influence points before World Congress becomes a thing. So let's keep it there. Assembly given discounts can stay too, it makes it easy to store up some points for World Congress in mid game when you still want to continue buying tiles.