This eyebrow mechanism shown is one of the simplest to do, yet very effective. It's basically a simple loop, not necessarily a circle, but a loop. One half is inside the puppet and operates as the control, and one half is on the outside, covered in whatever material you make the eyelids out of. The points on either side of the eye where the loop meets the head, act as the pivot point. When you pull down on the control inside, teh eyebrows on the outside, go up. When you push the controls up, the eyebrows go down. Like a teeter totter. This is the basis for most eyelid mechs, but some have more complicated shapes and configurations.
Great looking puppet by the way.