Basically, GIFs that are completely motionless (or nearly so) save for motion of the eyes. Generally, most of the motion is movement of light spots on the eyes so they they appear to be shimmering.
I've started using the tag animated eye gifs when I see them, but was curious it if they might need their own page, like Vibrating GIFs and Cinemagraphs More importantly, is there a proper name for them? I know there are plenty more GIFs out there of this, but I don't know what I should be looking under to find them.