For me it all depends on which group we're talking. I know this thread was likely started thinking strictly and solely about the X-Men, and not one of it's offshoots, but even then, with this movieverse, we have the original cast of characters and the "first class" of characters.
But again, it's all on which group, as follows:
X-Force/New Mutants: If they ever get around to expanding to this point, X-Force or the New Mutants seems like a great was to go. Even then it all depends on which incarnation we're talking about...
New Mutants: Let's start at the beginning. Say a New Mutants team in the 1990's (which would fit if Xavier brings Jean, Scott and Ororo together in 1983), or in the present timeline, which would allow supporting characters like Kitty, Colossus and Iceman to be instructors.
Some characters might be difficult due to what we've been given. For example, Sunspot (if you want to call him that) was shown (kind of) in the future in DOFP. Also, having Magik prior to Piotr is possible, but that relationship would need to be overlooked.
Anyway, the New Mutants would still be possible with great, unused characters, such as Cannonball, Cypher, Karma, Magma, Mirage, Boom Boom, Rictor, Wolfsbane, and Warlock.
So, which villain, if this was a strictly New Mutants story?
Selene would be perfect. Maybe an offshoot of the Nove Roma story. Maybe the Upstarts could be involved.
Another good option would be Warlock's story and the Phalanx. This story could end with both Cypher and Warlock dead, with Warlock's remains being sprinkled on the grave of Doug Ramsay. This could lead to Douglock.
X-Force Origins: Now they could start with the New Mutants (or at least a couple of them, like Boom Boom and Cannonball) and have them find themselves in some difficulty, at which point Cable, Domino and another (Shatterstar or Psylocke, for examples) would save them, and bring them together as the militant X-Force.
Emma Frost and the Hellions would be great, if Emma wasn't dead. Of course they could just play the "she made them think she was dead" card. This story could bring Trevor Fitzroy into the story, which would be interesting. And of course Fitzroy could lead into a story with the XSE and Bishop.
Of course any story with Cable as the leader could use Stryfe as the main villain. Just imagine the general audience's reaction when Stryfe's mask is removed. The Mutant Liberation Front could be included to expand on the villains. In addition, Deadpool could start off as a hired assassin, just as he was used in the comics.
Uncanny X-Force: Although I would prefer to see the New Mutants, or the X-Force Origins team (Liefeld's work), this is the team-up that most have speculated as to what we will get if (and that's a big IF) we ever do get an X-Force movie.
So if we had this line-up of Psylocke, Fantomex and Deadpool, etc., maybe the Apocalypse Solution could work. It's a very different storyline, and the main villain isn't really Apocalypse. Clan Akkaba and Ozymandias could be included. The idea of whether a child is guilty of his future sins is pretty intriguing.
X-Factor: Although I am not expecting to get an X-Factor movie anytime soon (if ever) the villains for this team should be different than what other teams would get.
Once again there are different incarnations of the team. To me there is the original X-Factor, which could be made up of some of the original cast, and the government group.
For the original X-Factor, in this movie-verse having some of the original members break-off likely wouldn't work, but the main stories would be Morlock Massacre and the Inferno story.
The Morlock Massace would be Mr. Sinister and the Marauders, though this story would make much more sense prior to an Apocalypse movie, Guess it could still be done after and could include Sabertooth and even Gambit.
The Inferno storyline would be ok, though the whole Limbo, N'astirh and S'ym villain pair would be tough to pull off. The Goblin Queen would look pretty good though.
Government X-Factor: I've always liked this version of X-Factor. It's a shame that it would be so tough in this movie-verse to bring Havok, Polaris, Strong Guy, Multiple Man, Forge, Wolfsbane and Quicksilver together, though I guess for most of them it would be possible. For me the difficulty is that Havok is the leader, which likely wouldn't work due to him being in the 1960's and 1970's timeline (unless that Havok was Uncle Havok - if you know what I mean)...
You've got to admit, Guido would be hillarious!
Anyway, Mr. Sinister and the Nasty Boys could work. The Mutant Liberation Front would also be a possibility with either Stryfe, or better yet, Reinfire as the leader. Maybe Cybur and Helle's Belles could work, but this one is iffy. Maybe Exodus and the Acolytes. Haven might work, but maybe she's too obscure for the GA.
ExCalibur: I'm not expecting this one ever, but if they were to take what I'm hoping will end up being a very popular Nightcrawler (after X-Men Apocalypse) and want to give him his own team, this would be the way to go. Not sure if this team could work without Kitty, but anyway, for Excalibur the best choices would be Arcade and the Crazy Gang, maybe, or Jammie Braddock would work, and of course, Ahab and the Hounds would make sense. Maybe Nimrod.
Then we have Wolverine:
We know they'll continue to make Wolvie movies, so who would be good?
Although I would have loved to have seen a good version of Lady Deadthstrike, I think that story is ruined. Same could be said for Ogun. Omega Red would be the next obvious choice. Cybur would be another interesting one.
X-Men: Some of the other options are:
Onslaught, the Brood (maybe that's too alien-ish), Nimrod, and Mojo (could introduce Longshot).
Here they are in order:
X-Men: Mr. Sinister and the Marauders - Mutant Massacre story
X-Factor Origins: Mr. Sinister and the Marauders - Mutant Massacre story
Wolverine: Omega Red
X-Force Origins: Emma Frost & the Hellions w/Fitzroy
X-Men: Onslaught
New Mutants: Phalanx
X-Factor: Mr. Sinister and the Nasty Boys
X-Factor: Stryfe and the Mutant Liberation Front
Government X-Force: Stryfe & the Mutant Liberation Front w/Deadpool
Government X-Factor: Reinfire and the Mutant Liberation Front
Government X-Factor: Exodus and the Acolytes
New Mutants: Selene & the Upstarts (Nove Roma)
X-Factor Origins: Limbo, N'astirh and S'ym (Inferno storyline)
Uncanny X-Force: Clan Akkaba and Ozymandias
ExCalibur: Ahab and the Hounds with Nimrod
X-Men: Mojo
X-Men: Brood
ExCalibur: Jammie Braddock
ExCalibur: Arcade and the Crazy Gang
So I guess what I want is:
#1. Mutant Massacre Story with Mister Sinister and the Marauders.
#2. Omega Red vs. Wolverine.
#3. X-Force (issue 100 cast) vs Emma Frost & the Hellions w/Fitzroy
#4. X-Men vs. Onslaught
#5. X-Men (or New Mutants) vs. Phalanx
#6. X-Men (or Government X-Factor or X-Force (issue 100 cast)) vs. Stryfe and the Mutant Liberation Front
#7. X-Men or Government X-Factor vs. Reinfire and the MLF
#8. X-Men or Government X-Factor vs. Exodus and the Alcolytes
#9. New Mutants vs. Selene and the Upstarts
#10. X-Factor Origins or X-Men in the Inferno storyline