Sidekick was a phone with full qwerty keyboard marketed to teens by T-Mobile in the US. It had data functions built-in such as AOL Instant Messenger, but didn't require a separate data plan. It was very popular with teens and the deaf community because it was easy to type on and messaging costs were very low.

Since she wants to play games, you want a smartphone. An iPhone will probably be the most satisfying device for this purpose, but there is a high up-front cost if you want to keep the monthly fees low.

You can play games on Android, of course, but the hardware capabilities vary greatly between devices. If you're talking about games like Words with Friends, anything will do. If you're talking about Temple Run or something else with a lot of 3D graphics, you need to make a more careful decision.