Possibly encouraging trade and movement of non-faction people, but how to safeguard such a system from abuse is not clear. If you have any ideas, put them forth.
When a player not belonging to the faction does some task in town AND leaves it safely town influence of controlling faction increases. Only AFTER the player leaves the town alive. Tasks - mining, using workbench to craft
good items (not some crap like knuckles), shoveling, doing quests, selling good items CRAFTED IN TOWN, buing bases, buing mercenaries, buing professions.
For example - a player comes to town, mines 10 ores (+10 pre-influece), shovels 5 shits (+1 pre-influence ), does shovel quest ( +10 pre-influence ), crafts 4 rifles (+20 pre-influence) and sells them to town merchant (+40 pre-influence). If the player leaves the town without dying, the faction in town gains 81 influence points. But if the player dies, all the pre-influence he/she collected is lost.
You have to be careful to choose tasks that can't be farmed easily. Like trading back and forth is not good, so I propose only selling items you crafted in the town (without earning pre-influence for buying them back and selling again). Also buying things via dialog, like beer/nuka cola would be good, if only it didn't add caps to bartender inventory (it does).
This should be generally small gain / player, so you would need many people to come to town to notice positive effect.