So first you need to determine which station is controlling - usually it is biggest one. Population within a system does not directly correlate to the influence algorithm, but it should be some margin easier for low-population systems.
If a faction gets a majority within a system they have a chance of taking the system over given that certain thresholds are met during a civil war. One of the thresholds include that minor stations and outposts owned by the controlling faction will need to be taken over first before the system's controlling station can be taken over. Another threshold is that the controlling station needs to be one of the stations in a civil war. A civil war is the only way for a faction to become the controlling faction for a system. The winning faction will be changing, or remain, the system's global faction to what the faction supports and changing the ownership of the controlling station.
As always, missions are the primary driver of influence for your faction of choice. Try to complete as many as you can - missions can be taken from any station in system but only for the faction that you want to increase influence.
Trades should be done only at station owned by faction you want to increase influence. Best is to take high supply items away and bring back high demand commodities.
Trading in illegal commodities increase the lockdown effect so should be done only at rival outposts or stations which have black market, because doing that will heighten security in the system which can lead to civil unrest and war.
Bounty hunting is not recommended. By removing the pirate threat, you are actively supporting controlling faction, therefore increasing their influence and doing the job of their security vessels.
The handing in bounty vouchers issued by controlling faction should be avoided. Bounty hunting should take place in nearby systems and Major faction (depending on your faction allegiance) bounties should be handed in at station that is controlled by faction that you support.
Destroying controlling faction ships including System Authority vessels. This will actively increase security and the chance of forcing the Lockdown and/or Civil Unrest state on controlling faction.
Miners should sell metals/minerals only at station which is controlled by your supported faction. This will increase their influence in the same way that trading does. Ideally you should look for metallic rings with pristine reserves.
Explorers should sell their exploration data only at
station which is controlled by your supported faction. This will increase their influence.