Pay attention, if you can, without batting an eye. That is, don’t play baseball with your eyelids. No, wait, that’s not right. Don’t flap your eyelids like bat wings…er…but that’s not how bats fly, as far as I know. What I mean is…well never mind that; let’s figure out what’s going on with “batting eyes.”
“Batting your eyes,” as you know, basically means blinking. In some contexts, it has to do with flirting: “When she began coyly batting her eyelashes at him…” It can also mean fluttering your eyelids, as you might if you got something in your eye. There are also older uses of the expression that have to do with sleeping (or failing to): “I didn’t bat an eyelid all night.”
The earliest definition of bat (at least this sort of bat) comes from an 1838 dictionary: “Bat, to wink.” A different dictionary, from 1879, narrowed it down a bit: “Bat, to wink, or rather to move the eyelids up and down quickly.” Notice that there’s no suggestion of flirtation there — that didn’t arise until the very late 1800s, and originated in the US. It was used in the turn-of-the-century (and aptly named) Century Illustrated Monthly Magazine: “You hol’ your head high; don’t you bat your eyes to please none of ’em.”
As to where it came from, when you “bat your eye” or eyelid, it’s got nothing to do with the kind of bat that’s a stick, nor the kind of bat that’s an animal. It has to do with the kind of bat that’s found inside words like “abate” or “bated breath.” That is, it’s the word “bate” slightly shortened.
“Bate” was borrowed from Old French, where it was “batre,” and had a couple of different meanings. In its original Latin form it meant to fight or strive — possibly including hitting. You’re probably thinking of “battering,” and you’re right. But it was also a term connected with falconry, and referred to the bird fluttering its wings: “The Gyrfalcon bated, and I came in to set him on the pearch” (1631). In 1673, John Dryden used this figuratively, to mean “restless”: “You are eager, and Baiting to be gone.”
Another meaning of “bate” was to reduce or bring down, as in this passage from 1460: “Theyre..wages be batyd.” And it’s sort of a combination of “bring down” and “flutter like a bird’s wings” that combined — along with bringing down the length of the word itself — that eventually produced the “bat” that you could do with your eyelids. And if you didn’t bat an eye during any of that, good for you; you’ve made it to the end.