Shut-eye is sleep. Informal, a little jokey, and a little old-fashioned — the kind of word you say at the end of a long day, not in a doctor's note. "I need some shut-eye" means you need to sleep. It paints a simple picture: you shut your eyes, and that is the rest. People often use it for a bit of sleep, a nap, or "some sleep tonight," though it can mean a whole night too. You would not tell a child "It's time for shut-eye" in a textbook way; you would say it among friends, or in a story, with a small smile.