We have here Sharpie retractable fine tip markers, which have the sort of tip I'm looking for, but a rather limited range of colors, and they still run about 2 bucks each.
Here's a classic, sharpie fine point. This may be the one, but I am a little concerned that tip is a bit to big for the spirograph. But these are only 99 cents if you buy them individually, and they come in about 30 colors. And you can get a set of 29 for $27.99. Sounds pretty tempting right now.