Currently, all minor factions provide pilots with access and rewards in proportion to the relationship with them. I think that can be extended beyond mere allegiance, to membership and rank.
Players should not ever rise to the stage where they own and direct the overall operations of factions, but their actions as members should hold sway outside the faction in proportion to their faction rank.
If someone high-ranking goes on a trade mission to another faction and assists them in taking care of a piracy problem, or in scaring off traders who are drying up their favoured runs, the relationship between the two factions should be positively affected.
Conversely, high-rank allies that don't follow the rules "when in Rome" should be treated as the public face of their faction, and their actions would negatively affect the overall relationship.
They should, in concert with other high-ranking members, even be able to trigger wars and build allegiances between factions.
(Inter-faction wars would probably have to be in-system only, I suspect, so as not to confuse the big war with all the little independent battles. Trade wars and general hostilities, on the other hand, could stretch some distance.)
A faction should provide rewards to allies similar to those enjoyed by their own members, based upon the rank of the pilot and the strength of the faction relationship.
Each faction should have a set of rules and customs that should be followed by members and visiting allies. Members that break the rules can be sanctioned or have their status revoked. Allies may have rules that conflict with the faction's own, and such differences would be respected, but larger transgressions would result in hostility.
Members should have the additional burden of responsibilities. These could be financial, like regular tithes, or something else, like community service work (anti-crime patrols, trade quotas to take care of shortages and surpluses, scaring off or permanently taking care of pirates or rival faction's traders, chaperoning valued partners across faction space, that kind of thing).
Allies would allow each other through faction space without scans or hassle from their own marauders (if applicable). This way, trade routes could be controlled somewhat. However, neutral traders would still have to be tolerated, as the factions would still need to be open for business, so lone wolf players could get by without any extra aggravation (beyond the usual).
Factions can expand into other territories at present, so that would allow the spread of a single faction separately from allegiance networks and power blocs. Also, perhaps factions might merge, if their interests and allegiances are strongly coupled?
All such increases in terriroty would attract the attention of the major faction forces, however, and that could be bad news for the minor factions - so any attempts at direct empire-building via mergars and acquisitions would have to be slow and steady. Increasing power via partnership, diplomacy and cooperation, however, would fly under the radar of the major factions for the most part.