Parents that use linux can easily have their computer play just one movie with the following script:
if [ "$1" ]
totem --fullscreen "$@" &
It starts playing the video(s) you want, but then prevents the kids from playing other videos or engaging in other unsupervised fiddling with the computer.
xtrlock needs to be installed, easily for example from the standard Ubuntu Software Center in Ubuntu 10.04.
You can create a Launcher in the panel for this script and a %F paramter, and drag videos to it from the File Browser.