1) The X-Men are weird, specifically, they're UNCANNY and/or EXTRAORDINARY. They are "the strangest superheroes of all!", as per their original tagline. Their stories have to be weird. Not like, Doom Patrol weird, but, still very weird. As an example: Doop. Doop is really fucking weird. He is the ultimate embodiment of comic book weirdness. Doop is a metatextual statement that says "this is a universe in which Doop can happen, so hold on". X-Men books are about, in part, how cool it is to be weird.
2) The X-Men are political. You consider their ethos, the time during which they were created, and who created them, and, the parallel to the civil rights struggle are there. Obviously there's way more to it than that, and you can't map the X-Men to any single minority group, but that doesn't mean the X-Men should be depoliticized. They are minority itself, they're anyone that's ever been discriminated against for any reason. They are about building a community around those shared experiences.
3) Corollary to that, the X-Men are part of culture. They aren't isolated from the world, they are a part of it, they're interacting with it, both in consuming it and in creating it. The world of the X-Men is one in which Jumbo Carnation from Morrison's New X-Men exists, and it is also one where Jubilee asks "Is it the shoes?", from Jim Lee's X-Men. It's true for the wider Marvel universe, but, lately, no line has seemed as isolated from the "real world" as the X-Men
4) Putting it all together, the X-Men are about us. Through metaphor and allegory, they're about our struggles to live in a society that hates, fears and discriminates. The Hellfire Club means something beyond "Chris Claremont got horny watching The Avengers (the tv show, not the comic)". So does Mojo, so does Magneto. It's true of all genre fiction, but of the X-Men especially. X-Men books need to speak to the human experience.
There's a bit more, but, those are things that I think of when I think of X-Men.