Hey... here's a neat trick for downloading a video from Facebook:
First, notice that your "Saved videos" can usually be found in the upper right menu (where birthdays usually show up) or under the "Saved" button in the left menu.
Once you click on the saved video, you'll find the file... click on that bad boy to play it in the browser.
Once it's playing, notice that there's no way to download it (that's what we're gonna "hack")
In the URL of the playing video, replace www with m (for mobile)
You should be brought to the funky looking mobile site (even though you're on a regular browser)
Play the video and then right click, you should see the following menu:
Save it as an MP4, crack a beer, and enjoy your Video whenever you want without having Facebook open.