Yep, degaussing is what you need. If I need to degauss a monitor that doesn't have that function built in (most old ones don't), I use an electromagnetic mass tape eraser that my father bought from Radio Shack in the late 1980s. It works very well. You have to turn it on and slowly pull it away from the screen.
This is good to know; it looks like tape erasers are a lot cheaper than degaussing coils on eBay. I'll find something!
I also can't help but think of this from Fight Club:

(And as an aside, I wonder how many kids just don't get this scene nowadays..)