I like to build an eye in the Vertex Modeler, duplicate it. remove a few polygons, then add thickness. This gives a shell that's the same inside diameter as the eyeball, then I boolean slice off as much as I need, copy it to another Vertex window, layer it to the eyeball, [in the Assembly Room] link them, make a blink, then repeat the keyframes as often as necessary. If you want the eye to blink on a more random cycle, push the keyframes around. An eye blinks roughly every second or two.

The picture of Beast is not his best, but the eyelids, I think, work fine.