Dumping about 5MCr-7MCr of exploration data will get you allied with a minor faction and increase it's influence because of the flow of money.
Charity missions improve the influence of the faction giving the mission.
Taking a mission then abandoning it knocks your reputation with the faction down and also knocks a little of their influence.
Lots of import/export trade has a positive effect on the faction that owns the station. You tend to find 2-3 stations in a system and a different faction in charge of 1 or more of them. If you're wanting to boost one faction at the expense of another, trade with that particular station, and destroy ships of opposing factions. Be careful you don't lock yourself out of the system with too much bounty against you.
Charity missions improve the influence of the faction giving the mission.
Taking a mission then abandoning it knocks your reputation with the faction down and also knocks a little of their influence.
Lots of import/export trade has a positive effect on the faction that owns the station. You tend to find 2-3 stations in a system and a different faction in charge of 1 or more of them. If you're wanting to boost one faction at the expense of another, trade with that particular station, and destroy ships of opposing factions. Be careful you don't lock yourself out of the system with too much bounty against you.