You're going to need some wood for the armature of the eyeball monster. I used scrap wood to make this, it could basically be made out of anything. To create the motion, I used a small oscillating fan. The eyeball would turn his eye from side to side. And you're also going to need something to dress you monster up with. I used some sheets.
Step 1: Build the armature...
How exactly you build it doesn't make all that much of a difference either. All of this will be covered up, so as long as the frame holds up, it will work. An important feature of my design is to put a flat piece of wood at the top of the frame, where the neck would appear to be. This is where the fan will go, so it should have a stable place to sit. Build your armature accordingly based on how your fan will fit into the frame.
I next added the fan to the frame. The fan should be resting on some stable surface. On my armature, it's a flat piece of wood. I secured the fan by taking 4 long wood screws and drilling them through the frame so they would come out just above the base of the fan, without going into the fan. This allowed the fan to be easily removed by unscrewing the screws, but it also kept the fan attached to the frame pretty well.
I based the armature off of the armature that I had already made for my Grim Reaper Animatronic, so the two are very similar.