You know, I haven't personally seen this problem before, but I've heard of it from someone else just recently. I'm not sure what it is, but I'd start by checking to ensure the power supply is working properly. The //c kind of has two power supplies - one is the external brick-on-a-leash that supplies 12v to the internal power regulator, and then the internal power regulator circuit converts that to the 5v (and others) needed in the computer. Check the voltage inside the computer at one of the chips, and be sure you are actually getting a clean 5v.
Another possibility is a cracked solder joint in the video section - try wiggling everything. On the IIe and other machines, there is a video level pot that causes weird things if it gets dirty. If the //c has it, try pressing on it.
Of course, this all assumes that the problem is in the //c, not the monitor - but I'm sure you tried that already :D
-Ian