Restarting it did nothing. I then tried zapping the PRAM (by pressing and holding the Command-Option-P-R keys when booting) a few times, and that didn't help.
I booted from the OSX install CD, and still nothing...
At that part, I started to freak out -- and wondered if the MacBook itself was dead. So I booted in FireWire target disk mode (see http://support.apple.com/kb/HT1661 ) and connected it to an iMac, and was able to get to the drive without any problems. I backed up everything that was important, worried that the drive might be the next thing to go.
At that point, I figured that I was probably going to need to purchase a new MacBook Pro. But I decided to try one last thing...
I rebooted the MacBook one more time, and zapped the PRAM, and... Voila! The video was restored.
I'm still not sure if zapping the PRAM an additional time resolved the issue, or if booting in target disk mode is what did the trick. But if you find yourself in the position that I was in, give this a try.
UPDATE: After another10 hours or so of the MacBook working normally, the video went out again. My "target disk mode / PRAM zap" trick didn't work. It appears that the video comes and goes, with no rhyme or reason.