This is a great question (even if it's kinda off topic).
There are so many different games on mobile devices (even apps on Facebook) for kids, it's really hard to know which are the ones you want to allow and which to refuse. A good way to differentiate is to read the ratings and reviews within the App store, and determine whether the game has the elements you are looking for your child. Some games have great educational value for kids, while others are pretty much refined sugar for the brain. Definitely worth checking it out, becuase there is a lot out there.
For me, I love letting my kid play with games like Video Touch or Sound Touch, whch work with animals and objects and then moved onto games which work with shapes and colours to assist with their learning. There are also some great reading apps for kids that really do a lot, as well as fun sing-a-long apps that are fun for the whole family.
Have fun checking it all out :)