Who's asking: Jennifer Jordan, Milwaukee, WI
Jen is grossed out by this expression, but I am sorry to say that it's a kinder, gentler version of the earlier "Keep your eyes skinned." Both phrases turned up in American popular fiction and news reports in the mid-1800s; "eyes skinned" dates back to 1833, while "eyes peeled" shows up in the 1850s. Both expressions suggest that a watcher should physically hold his eyelids open, in order to stay alert.
It is possible to sleep with one's eyes open, though usually not with one's eyes wide open. Some people's eyes don't close completely during sleep, which is a sign of a neurological problem. Sleeping with one's eyes open is called lagophthalmos, which means incomplete eyelid closure. It can happen when you're intoxicated or on a powerful sedative, or in people with thyroid disorders. It's not normal and it's not good for your eyes, so if you find yourself sleeping with your eyes open, consult a doctor.
Earlier this year I did a good bit of research for a client who is writing a series of poems with the central image of a glass eye, so I now know more about the anatomy of the eye and the mechanics of vision than any lay person should. Eyes do have their own transparent skin, the conjunctiva; peeling the conjunctiva off one's eyes is not a good idea. I'd go into detail, but is that necessary?