Co-op in the wasteland, Bethesda. We've made a mod for this in Fallout 3 (and partially in Fallout: New Vegas). It can be completely optional, and players can just let a friend tag along (like in Saints Row games). Seriously, this is even simpler than Assassin's Creed: Syndicate's ability to support multiplayer modes.
You could trade resources between each other, including better gear your partner may have unlocked, like in Borderlands. Also like Borderlands, the combat difficulty could scale for the increase in active players. It would be especially ideal for those who don't normally play hardcore RPGs like Fallout, as their more experienced friends could help them survive the Wasteland.
The most interesting potential, though, is if players have different faction allegiances, and they use these to make new quest branches that otherwise wouldn't be possible. Imagine you and a friend igniting a war, or preventing one, by using your various connections and allegiances. The potential in the game's sandbox is very enticing.
No one really wants a competitive Fallout game, and I honestly can understand that. The series has never been about competition, but survival and personal stories. So let us build stories together, Bethesda, and tear down the wall preventing friends from playing with each other.