No camera is as cheap as the one you already have built into your phone. It is however tiresome mount and protect it for making videos of a tour.
Next up, most modern digital cameras have a video facility built in. And good ones like my Olympus 720-D come with a built-in tripod mount, which makes mounting the thing stably to the handlebars is a doddle. I bought it for 100 Euro at Lidl, the supermarket. You can charge it without taking out the battery but it cannot charge and take a video simultaneously. Controls too fiddly to be handled while wearing full-finger gloves.
If you want a dedicated cheap sport video camera for the bike, I can recommend the Kodak zX1 which has a built-in tripod mount, uses two rechargeable AA batteries, and can be fitted into an optional custom thick rubber case to protect it against sporting knocks. Its primary purpose is video, though it will also take stills. Not a high quality camera but then they spent the money on a functional camera, not a fragile prize-winner. Mine is pretty old and still looks good in the optional rubber case despite some serious knocks, so I think the rubber case is a must for sporting use.
