Possible Clues:

• Mobile cover, essential to winking and blinking!
• Guard for pupil that's batted sometimes
• Twisting in grip of crocodile, yelling 'It's snapped shut!'
• Lashes grow on it
• Shadow canvas
• Site of many lashes
• Optical shutter?
• If it's closed, you can't see priest you once introduced to daughter
• The blinking thing!
• Facial shutter