They started with X. If they did that with hurricane names we'd have a hell of a lot better weather I'll tell you whut.
The idea was that they had "baby boomer" after the "greatest generation." For some reason they divided out something called the "silent generation" in between there. I guess you had to shtup right after the war to make a baby boomer, so there was some fooling around by Jodie and your girl back home that they didn't like to talk about much, resulting in the silent generation. Makes sense. Anyway, there was the Lost Generation before the Greatests, and the Losts were already a thing way before the "Baby Boomers" started the living-memory conversation about generations, unless you count the perennial harangues on streetcorners about every generation being "ye generation of vipers."
The idea of Generation X was that they were all soooooo disaffected by like, everything, man, that, other than watching Anthony Michael Hall trying to get some and watching Molly Ringwald invent Resting Unpleasantyounglady Face, they just. didn't. care. There were problems but why bother solving them, nothing works, no parties, no government, business is just as bad, why vote, why care, there's too few of us, Boomers got all the good jobs, whaaahhhhhhh. That was the (pretty much self-chosen) stereotype. At least it was self-chosen by the guys writing books about their generation. A guy named Douglas Coupland wrote the book whose title gave us the name.
Googling around, I just found that on Robert Capa tried to use "Generation X" on "the young adults of the 1950s," so I guess those were the Silent Generation," but it didn't stick.
But once it did stick, there was always a solid lobby for calling Millennials Gen. Y, landing us on Z now (it's also convenient that Max Brooks set "World War Z" in the then-near-future, and has a character say something like "Yeah, I was in Generation Z," which is funny because it was also the generation coming of age when the Zombie apocalypse came.)
It appears that people want to use the greek alphabet after gen z, and we'd better hope the dumbass brave-new-world, zoo-behavior fad is over by then, because that makes the next batch Generation Alpha.