Do you have a old phone? Most phones from the last couple of years will have better cameras and features then any 200$ handheld camera you can buy now. There are several android apps that can turn your cell phone into a web cam too.
If you don't have a older cell phone, I would suggest looking into GoPro. For altenatives, check the Contour 2 and the Polaroid Cube. Some questions:
1. Much much storage and what kind of storage do you prefer? Onboard vs usb vs SD vs micro-SD
2. Do you need to have Wifi?
3. Does it need to be waterproof? Indoor or outdoor?
4. How will you be recording? Mounting on you? Something else? Point and click? Remote?