It has happened to me as well. The thing is, it still has to be raining after the evolution. Should it not be raining anymore your Sliggoo will not evolve. I would recommend waiting for awhile around a route where rain is usually present such a route 19.
So don't worry, there is nothing wrong with your game or your Sliggoo. It's just the way the mechanics work so just consider yourself unlucky in this situation that it had to stop raining. It just happens randomly. For some people evolving Sliggoo can be a hassle for the simple fact that the rain keeps fading away right after the battle, so if t still doesn't work after you have tried it again, I would recommend you just leveling it up in rain with a Rare Candy. However save before you do so, just in case luck strikes you again and it stopped raining. Like I said above, I recommend doing this on route 14 or 19. It tends to rain a lot there.
