Whenever writers leave, they also leave the specific things they had in mind for their characters. I always think of Louise Simonson's Rusty and Skids, or Joss Whedon's Armor or Joe Kelly's "All New, All Different Danger Room X-Men" or Lobdells' Cecilia Reyes or Ann Nocenti's Longshot.
Creators seem to have a knack for creating new Mutants, but never seem to figure out what to do with them "between stories" or after they finish the arc they wanted to write. X-Men aren't archetypes...not in the same way that, say, Captain America is an archetype. There's no definitive Mutant character.
I always thought doing away with Grant Morrison's idea of a "Mutant Subculture" was a bad idea, if only because you could use many of these mutants as background characters in the way that characters from WildCARDs shifts from background characters being foreground characters and vice versa.
If anything, Peter David does this the best with characters he seemingly pulled out of a punch bowl to write about. His strength is figuring out how to foreground and background characters in a serialized manner so that the reader understands that the narrative is more like a telescope that focuses in on one character and we see the world from their eyes in an arc, then switches that telescope to another character in the next arc.
If I were plotting the next group of X-Stories, I'd probably do the following:
1) Brian Wood on Generation X -> What Wood excels at is these "slice of life" stories with X-Characters. One of the best Generation X arcs is when he separated the characters and had them interact with different facets of New York. I'd love for him to do that with more characters across the globe. Maybe you have one or two characters that are the same, but a Generation X book where we follow Jubilee traveling the world to meet different mutants (that are all older characters in new contexts) would be interesting.
2) Joe Casey on the X-Factor -> I'd put Lorna Dane in charge of Serval Industries with 'Berto DeCosta. I like the idea of Corporate Superheroes, but one that would be in charge of 1) doing good for a large group of people 2) putting a good face on corporations. And I'd resurrect the idea of a "sustainable source of power" created by mutants to run Serval Industries, with the daughter of Magneto as the head of this corporation. Doing corporate good, but with the notion that corporate good may not be good for individual good, would make for an interesting title.
3) Cristos Gage on the Hellfire Club -> A story about the movers and shakers of mutants, with an eye towards a long history that goes back to the External, sort of like how SHIELD goes back further than the founding of the organization. I'd love for an X-Book that would do the deep dive into the continuity of the series as it relates to current day situations. I'd love for Hellfire Club to be this rotating cast Illuminati-esque characters that are bound by being Mutants to solve Mutant problems. Anyone could be a Hellfire Club member. All you have to do is answer the call.
I don't know. I put those ideas down 30 minutes ago and they still feel like the problem that I mentioned above -> X-Characters only seem to exist for writers, and then go away once writers jump on the title and want to do their own version of "Kitty Pryde" or "Wolverine", or take on ersatz "Kitty Prydes" and "Wolverines", or create combinations of those characters they liked as kids. Deadpool is, after all, "What if Spider-Man mixed with Wolverine" and Cable is "What if Punisher had ALL the guns."
I think what made the X-Men special was that Giant Sized X-Men took away all the old Mutants and gave us new Mutants. I think the only way you could have a newish team is to do what Claremont did: wipe the slate clean of all the First and Second Generation teams, put your new team in place (made up of old and new) and then slowly put back the First and Second over the course of many years.
That's how the Bendis Avengers worked really well initially. People had all new avengers and waited to see what happened to all the older ones that eventually came back into the books.
Could you do that to the X-Books? Wipe the slate clean with the intended rule that you could not feature any of the First Generation X-Men (Cyclops and the Crew) or Second Generation (Storm's Team up through to the X-Genesis changeover with the Adjetiveless X-Men and the first X-Factor). The rule would be, "If Claremont wrote them on his first run of X-Men, they cannot be in a title for at least two years."
Would that work to refreshen the X-line? Almost a clean reboot that wipes the decks of the 1970s-1990s X-Men, leaving it open for a lot of the secondary characters to be in the forefront. What would those teams look like?