You call them "local" factions, but I prefer "minor" factions, since they can have a presence in multiple systems.
Benefits include:
- You get friendly/allied with the major factions by becoming friendly/allied with multiple of the minor factions allied with them. Being friendly/allied with major factions makes you automatically friendly with all of their minor factions, and has other benefits.
- Lower fuel and repair costs at any station they control.
- More/Better missions on the bulletin board. If you look at the bulletin board, you can see that missions require a minimum reputation with the faction posting the mission, and some require "friendly" or "allied".
- In a few cases, it's how you gain a permit to a system. There's a bunch of system permits tied to military rank with the Federation and Empire, but there's also systems controlled by minor factions that you have to get friendly or allied with to get the permit.
- Less attention from their system authority (police) ships. Less likely to be interdicted, slower to get scanned on your way into a station, etc. This makes travel in that system easier, and especially makes smuggling easier.
- It's further from "Unfriendly" and "Hostile". Those have definite disadvantages, and you can get away with more things before you get there if you start at friendly or allied.
- Their ships show up green on your scanner. Depending on what you're doing, this can be helpful. I find bounty hunting in a friendly/allied system easier, since I can skip scanning green ships.
- You get nicer messages from the station when you request docking.
If you rank up from "Friendly" to "Allied", it's basically just more of the same types of things.