It's a bit remarkable how much you can do with things that are just lying around—especially when they're Legos and electronics parts! The only parts we purchased for this were some wooden tongue depressers ("craft sticks"?) and permanent markers.
The egg is held up between two rubber tires that have just the right combination of sturdiness and compliance (there is actually another ring-type tire stuff inside them so the contact surface is smaller, otherwise they swallowed the eggs). The whole structure is made of Lego parts from one kit I had gotten as a gift (plus a gear that I had to add), and the rest is just wooden sticks, hot glue and rubber bands!
The pen is mounted on a servo for moving back and forth, and its weight keeps it down on the surface of the egg. So far the pen just drags along, so it's not like this could draw anything complex (plus the resolution on the servo leaves something to be desired). Still, I've got no complaints so far.
The stepper motor is driven by a simple circuit made from a bunch of transistors that just selectively ground two of the four poles to the motor. It's very simple and I doubt it's original (I don't really remember when I drew the diagram the first time) but it works pretty well.
Right now the egg just spins steadily and the pen can be controlled manually, but that will change soon!