There are a few things to consider when buying anything for
a teenager. First and foremost, teenagers break things -- including
Digital Cameras. You want to find a camera that fits into a back pocket or a purse side pocket but won't get demolished in the process.
Samsung MV900F

The real memories your teen wants to capture involve taking pictures of themselves at someplace awesome, standing alongside their BFFs as they all pose in an appropriately silly, ironic, or bro-tacular fashion. On most cameras, self-portraits involve a tricky trial and error process, but not so with the Samsung MV900F
Nikon Coolpix S30

Here's a super-cheap lifeproof camera from Nikon, the S30. It's got you covered on small drops (up to 2.6 feet), and you can even use it under water (down to 9.8 feet or 3 meters). It doesn't have the best optics, or quickest autofocus, but the S30 is nicely appointed with 10-megapixels, a 3x optical zoom, and 720p HD video.
Sony Cyber-Shot DSC-W650

For most teens, the smaller the gadget, the better. The Sony DSC-W650 is a sleek and simple ultra-compact digicam. Sporting a 16-Megapixel Super HAD CCD Sensor, BIONZ image processor, 5x optical Carl Zeiss lens with optical Steady Shot Image Stabilization, 720p HD video capture and Sony's Sweep Panorama shooting mode