I can only talk about games that i played elsewhere and i know they have mobile versions:
The Room -- cool looking puzzle game. There are three of them already, i only played first because it's the only one on PC.
Ace Attorney Trilogy and
Ace Attorney 6 -- are on mobile and are easy recommendations, unless you're some kind of purist. I think first chapters are free to play on these versions and then you buy the rest of them.
Super Hexagon is on mobile -- arcade style game when you have to avoid walls by rotating a little triangle around. Check
free flash version to see if you're up for it
Jelly no Puzzle has android version. The best designed and hardest puzzle game i played. Again, you can check out free PC or
web version to check if you're up for it.
Super Crate Box is on PC and iOS. Again, it's freeware on PC (even has Steam version).
Desktop Dungeons -- brilliantly designed "rougelike" game (comes from way, WAY before that term got stained by flood of too similar games) has iOS and Android versions as well.
There are few more i am forgetting -- particularly one flash game where you control a knight that can only go left or right...