I don't know how difficult this project would be, as I am an Arduino newbie myself. You can read sensors easily with an Arduino, but I don't know how you would log it. I'm sure there is a way.
As for the whether using an Arduino is the best approach, I know (from a friend) that there is an app on the iPhone that measures your sleep pattern - and shows you how much time you spent in REM, deep sleep, etc. It uses the iPhone's built in accelerometer to record this information, so it is possible.
If you wanted to do this using a video camera, you could illuminate the subject using infra-red LEDs, which won't disturb the subject.