If you are trying to write a sequel and you find you can't think of anything surprising to do with the set-up as it is - I don't really think a reboot is going to help you that much. The big issue is that, after rebooting the series are you really going to take all those big narrative risks that you couldn't previously. So we reboot Fallout, you go down into the vault for the beginning of the nuclear war, sleep for 500 years and when you come out, what? There was no nuclear war and humanity is getting along fine with their flying cars and vastly increased life spans - no you're still going to be in a post-apocalyptic world and the same basic set-up is going to be the same. Brotherhood of Steel getting boring - just set the game somewhere where they are not active, or 10 years later when they have had an internal civil war. Deathclaws getting boring, introduce some new genetic mutations so they look completely different. Doom, you can make any darn demons you want, because hell ain't a small place to be. (Actually how about not rebooting it, but not setting the next one on a generic derelict space colony)
I feel like there should be a (obviously creativity-stifling, proto-facist) law, saying that you are only allowed to reboot a property after it has been on hiatus for at least a decade. Want to re-imagine Battlestar Galactica for the 2000s, great go for it. You've not made a Superman movie since Christopher Reeves' tragic accident, great, go in any direction you want. You've billed Mass Effect 3 as the ultimate conclusion to an epic saga and now you've written yourself into a corner, but you still want to churn them out every few years? Nope, naff off and make something original for a decade and we'll talk when generation 11 roles around. You made a great game called Mirror's Edge, and now the fans want more, but you don't want to call the game a sequel for reasons I'm intensely perplexed about and can't think of any scenario that might involve Faith having to run around on rooftops after the ending of the previous game...wait, really? What are you thinking, it's fine as it is.