This would be great if you wanted to talk to yourself without anyone understanding what you’re saying, for example, or if you didn’t want anybody to know what you were writing down. It might also be useful if you enjoy lots of foreign language media (see music, tv shows, movies) and want to experience the “original” rather than through dubs and subs.
Personally, I would go for either Russian or Estonian, both of which are because I frequently listen to lots of music that are in those languages, and they aren’t spoken by any of my friends.
Tokein’s elvish, or other language I think he constructed several unique languages just for his backstory.
His cousin and him made their own language growing up they used to talk to each other, so it didn’t start with his books.