Levi Burner, who has written several article for DIYTripods, wrote in with an excellent DIY Camera Gimbal setup, shown above with a GoPro. Although it could be used with a quadcopter, it should work equally well as a handheld device.
Details of the build can be found here. Below is (part of) a rough sketch diagramming how this setup will fit together. Although it reminds me of sketches I sometimes do before building something, my artistic ability isn’t quite up to Oscar’s standards. Fortunately, I’m pretty good with AutoCAD (or Draftsight for my home projects).
If you’re wondering how it stacks up to an unstabilized camera, check out around 0:35 in the video after the “read more” thing. There still seems to be some movement, but the results are much better.