Well let me give you as an example what I have been doing today over the course of a few hours:
I went to a system called "Ho Hsi". In this system,like in every system, there are 4 internal system factions sharing influence/control over it. The faction currently in control with 47% influence is the "Ho Hsi Corp" an independant faction. Directly after that with 46% is "Jet Comms Limited" a faction allied with the Federation.
Because Ho Hsi Corp is independant and controls the system, the system itself is independant too.
I decided that I would make Ho Hsi my current "home" and I want to give Jet Comms Limited the edge in influence to eventually start a civil war and take over the system for the Federation.
To do this, I would only accept missions issued by Jet Comms, do these missions, thereby raising my reputation with both them and the Federation and also slowly increasing their influence. Every other commander, whether in solo mode, private group or open play who does the same thing will also add to their influence. Of course other commanders might do the opposite and reduce Jet Comms influence or increase Ho Hsi Corps, but that's life!
i'm currently trying to rally people on the Internet to help me make Jet Comms the main faction in Ho Hsi.
So far their influence has risen by 0.1% and I am now "friendly" with them, meaning all their ships and stations are green and allied with me, helping me out in a fight if they see me in one.
So I, for myself, am trying to be the "hero" of Jet Comms Limited.
Will I succeed? Who knows! Will the game acknowledge me? Preeeeeeeetty sure not one bit. But hey, it's a living!
edit: To show that what I am trying to do,especially the rallying of people is possible, check out this link:
http://www.reddit.com/r/EliteDangerous/comments/2p2nbm/player_event_the_crimson_revolution_in_lugh/