For gaming purposes, I also have to recommend Manjaro purely from how great it handles installing graphics drivers. I wouldn't necessarily recommend it for someone's first Linux install, but once you know the basics in case something breaks it provides a better gaming experience out of the box.
I wouldn't be surprised if Mint and Ubuntu are both in a better state in that regard since I last used them. Last time I used Mint on my gaming rig, the bundled Mint drivers had something funky with them. I don't remember what it was but I do remember I had to reinstall them. This was around Ubuntu's 15.04 I believe?
