The easiest solution would be to use an Android HDMI stick connected via an HDMI or HDMI-to-DVI cable. You can make a short video, install an appropriate video player app, and have it loop. Of course, these HDMI sticks will only work if your monitor has an HDMI or DVI input. If you need VGA or DisplayPort, you'll need to look at other similar solutions.
Alternatively, you could build your own Raspberry Pi-based system, use a Roku or Apple TV (or any other "home media player" device, etc.
Yet another option would be to create a video with your desired content, burn it to a DVD, and hook the monitor up to a Blu-Ray or DVD player (again, assuming you have the appropriate cable to connect compatible ports from each device).