The end mounting assembly should be sturdy enough that you can attach small video cameras to it as well -- anything that uses a standard 1/4" tripod mount.
There are a few bicycle camera mount Instructables already out there -- kudos especially to MikeIsOrganic's for the inspiration for this one. However, most tend to require unscrewing the camera if you want an unmounted shot, which can get tedious and might cause you to miss those transient or candid shots!
Step 1: Parts and Tools
Tripod ball head with quick release plate -- $20 on ebay
3/8" x 2" Bolt for mounting tripod head -- $3.50
3/8" Nylon lock nut -- $1.00 for two
3/8" Washer -- $0.09
1/2" Electrical conduit hanger -- $0.50
8-32 x 1.5" Machine screw with nut -- $1.00 for eight
Total Cost: $26.09
Electric drill with suitably-sized bits for the bolt and machine screw
Wrench and screwdriver
Duct tape, electrical tape, or thin flexible rubber (optional)
The tripod head is really up to you -- I'm quite happy with the DynaTran ATH-02H (ebay search) I found -- but most anything should work. The ATH-02H has a single lock lever, with full 360 degree spin capability, as well as 90 degree tilting in either direction, and the all-important quick release plate. It's also possible to remove and attach just the quick release plate, if you don't care about the articulation (which would reduce the overall height of the assembly). Whichever tripod head you choose, just make sure you match the size of the mounting bolt/nut/washer combo to the size of the tripod's mounting hole (remember to check the thread pitch too!).
The electrical conduit hanger I used can be found at Home Depot. Alternatively, if you can find a cheap throwaway bike light, you might be able to use its mounting bracket instead. The key is finding something that fits your bike's handlebar tube. Note that the conduit hangers at Home Depot labeled as 1/2" actually snugly fit a 3/4" tube such as the one on my handlebars.