The first Nancy Drew game is labeled as a "mystery role-playing game", which is interesting, because it doesn't have any traditional RPG elements whatsoever. There are no battles or menu navigation.
What would a mystery RPG look like? I think it'd be a game with RPG segments where you fight bad guys, and mystery segments, where you get plot info and question suspects. It'd be a lot like the "Death on the Nile" game I discussed yesterday, only it has RPG segments in between the story chunks, not hidden objects segments.
I'm told "Sakura Wars" works just like that. The games alternate between visual novel/story sequences and RPG battles. Although at this point, the series is really heavy on the story, and very light on the RPG stuff. If the story was a mystery, then yeah, I'd called that a mystery RPG!
Another possibility would be to have mystery-based RPG battles. Like, instead of "attack" or "defend", you'd have options like "search" and "investigate". That sounds like it could be complicated, but I'd be very amused to see "Nancy Drew used interrogation on suspect! It's super effective!". Perhaps we'll see something like that in the upcoming "Detective Pikachu" game?