i'm in the beginning stages of trying to flip control of System and main station right now. belongs to one of the larger player groups. well, prior to the Live/Legacy split.
while i am working by myself it is a smallish population, so possible to do in a relatively short amount of time. the larger the population of the System, the more work you have to do to change the minor factions' influence.
as others have stated, run missions for the faction you want to put in charge and don't run any for the faction you are trying to displace. it's fun to watch the Influence percentages change everyday.