This is sorta a loophole to making an Underdark PC race. Also, I'm a little concerned about the direction "Underdark Ranger" is going to go with that player of yours, will they also be a dual-scimitar-wielding Drow?
I like the idea that the player characters are not supposed to be familiar with the Underdark. That doesn't mean they can't be Underdark races, they just need a reason to not be very familiar with it, so I'd say no to getting to pick Underdark as a favored terrain. I'd also be cautious of if they ask for "caves", but I could see that working as the Underdark both is, and isn't a cave in the sense that surfacers deal with an interact with them.
Either way, I think any sort of familiarity with the Underdark is going to, by-and-large, create a lot of "rulings" situations where you're going to have to determine if the player's abilities are actually any use at all. I think the worst one is going to be the "can't get lost" in favored terrain.