So, courtesy of the Mount Vesuvius screenshot we know influence will be a part of this game. I figure we could start speculating on how it would be used, either in old ways or new ways.

1. Diplomacy. I love how influence costs for even things like going to war not only keeps things balanced for multiplayer and single player, it also discourages warmongers. One way this could be improved even further is that maintaining a state of war (or perhaps even a state of peace in some cases?) requires an influence maintenance cost that goes up over time, either combining with the stability mechanic or causing stability to go down if you can't pay. Also, perhaps you could trade influence? Like, if you are running low on influence to keep your war effort up you could give a friend a tech in exchange for some of their influence, which would basically be that civ publicly supporting you and your war effort.
2. Territory expansion. Since we know outpost type expansion is going to be a thing, maybe influence can be used to speed the process up, either through encouraging any natives to join up or telling your people all the great opportunities they'll have there. Maybe if it's close enough you can spend influence to try to steal other colonies out from under the creator, or perhaps make some propoganda to decrease stability in a nearby region so you can take the city over more easily? It could also be used to convince city states to join you.
3. Archaeology. We've heard you can dig stuff up from the ruins of your old cities, perhaps if they were orignally yours you get influence and if they belonged to someone else (you thief) you can trade them back to the owner.
4. Government. There are many ways they could go with this, but the system of nation vs world, freedom vs order, etc. that they have going means I suspect it is more like ES2's government system, so I shall focus on that. The main issue with this system is you never had a reason to change types of government and you are unlikely to have "parties" in the early eras if at all, but the culture switching mechanic could arguably already serve this function (it's safe to assume England's a monarchy of some sort while America's a democratic republic). Influence could be used as both an upkeep cost for whatever the equivalent of laws is and/or a cost for switching government types. Perhaps instead of one "party" you'll instead have mutually exclusive types of law you can use with the more towards the end of a spectrum you are the more advanced (and expensive) laws you can get. The main issue I see is that it doesn't exactly encourage centrists, but I guess those people rarely become famous anyway . . .
5. "Culture" tree? We don't know what that middle button in the lower left corner does, it could be a culture tree ala Civ6, although that would make it hard to "save up" Influence like in previous games.
6. Espionage. Spend it to plant the spies and have them conduct certain missions.

Well, I'm out of ideas for now. Any thoughts?

Maybe it could also be used to represent propaganda during the late game, literally influencing public opinion. 

During peacetime, this could be used at home with the politics system like you said, allowing for otherwise unpopular policies/ laws/ whatever without causing decreased stability. Abroad it could be used like the ideology system in civ 5 or the recruit partisans mission in civ 6 to spread dissent for an enemy government type. During wartime, it could help boost morale, or hurt enemy morale, which would give a benefit to troops and allow influence to be put towards the war effort.