Rainbows End is the name of the retirement community where Lena Gu and Xiu Xiang live. As Robert notices, there's a huge typo in that name. The developers probably meant to call the retirement community "Rainbow's End"—note the apostrophe. That would mean something like, "Oh, what a wonderful peaceful place we've discovered here at the end of the rainbow." Alternately, they might find a pot of gold at the end of the rainbow, or at least some Lucky Charms.
But without the apostrophe, what it says is more like, "Rainbows, like all good things, eventually stop." Now, would you want to live in a retirement community called "Rainbows stop eventually"? That makes it sound, well, like a place where people go to die. As we hear about Robert, "He'd never been able to decide if that spelling was the work of an everyday illiterate or someone who really understood the place" (14.100).
So… why name the book that? From Robert's perspective, maybe the world looks at first like everything good is or has ended: his ex-wife is dead (and for all his meanness to her, maybe he did have some affection for her); his poetry is gone; heck, even the books he loved are gone, even in the UCSD library. So "Rainbows End" kind of fits for Robert's first experience of this world—everything he loved is gone.
But that's not how the book ends. By the end of the book, Robert has found pleasure in his new math/science work; he's learned that his ex-wife is alive (and he gets a chance to make it up to her and to the rest of his family); and, yeah, the library has no books, but it's still a cool place to be.
So maybe we should read "Rainbows End" as meaning something like, "rainbows end, but then other rainbows start—it's not the end of the world."