An eyelid was a flap of tissue that protected an individual's eyes from irritants and light.
Vulcans were known to have an inner eyelid which protected their eyes from the harsh sunlight on planet Vulcan. While traversing Vulcan's Forge, T'Pol assured Jonathan Archer that hers would protect her from the sun's glare. (ENT: "The Forge")
In 2267, on a mission to the Deneva colony, Spock's inner eyelids enabled him to completely recover his sight after being blinded by a bright light dose of 1,000,000 candles per square inch for several seconds, intended to free his nervous system from a flying parasite. (TOS: "Operation -- Annihilate!")
Xaheans had interior eyelids to the left and right of their eyes, invisible unless they blinked. (ST: "Runaway"; PIC: "Remembrance")
In 2368, Geordi La Forge remarked to Hannah Bates that he felt as though his eyelids had lead weights attached. (TNG: "The Masterpiece Society")
In 2372, when Jake Sisko tried to awaken his grandfather from a nap, the older man facetiously asserted that he was not sleeping, just checking his eyelids for holes. (DS9: "Homefront")
Displacement of the eyelids was a possible symptom of Tuvan Syndrome. In 2375, Doctor Julian Bashir told Luther Sloan that Koval had this symptom. (DS9: "Inter Arma Enim Silent Leges")