AFAIK there is no RAW way to gain new skills. Like, there's probably mention of it somewhere in the DMG in an offhand "you can also reward your players with new skill proficiencies," but I don't think there's any explicit mechanic for it.

At the table I play at, it's fairly accepted to do little changes to characters during downtime. In this case, dropping one of your current proficiencies to pick up Slight of Hand would very straightforward and the easiest way to handle this.

If I were DM'ing, I'd ask the player to make a little narration out of it - how do they gain this proficiency? In the case of a rogue, maybe they fall in with a crew for one job, and there's some delicate operation they have to do that involves slight of hand. I would then call for a couple of skill checks to determine success (get some gold, maybe get arrested briefly xD). And at the end you have your slight of hand proficiency.