How to create eyes idle movement?

Ok I didn’t see what you were trying to randomize at first. Realistic movement would be more something like that then :

My curve goes from 0 to 1, 1 means eyelids opened. It closes then opens immediatly (I’ve set the keys on smooth, but you can try linear as well)

Then use the random delay like you did :

You can also randomize a bit the timeline play rate :