The other day a friend asked me about the origins of the phrase “to bat an eyelid”, which is normally used in the negative – he didn’t bat an eyelid at the pink elephant in the fridge – and means that you don’t react or show emotion when surprised or shocked. Or in other words, you took it in your stride. We wondered way it’s ‘bat’, which seems a strange thing to do with your eyelids.
The same verb is used in the phrase “to bat ones eyes/eyelashes”, meaning to open and close your eyes very quickly several times, intending to be attractive to someone [source].
According to the OED, the verb to bat is a variant of bate (to flutter as a hawk), from the Old French batre (to contend, fight, strive, flutter), from the late Latin batĕre/battĕre, from the classical Latin batuĕre (to hit, beat, pound). This comes from the Proto-Indo-European prefix bhau- (to hit) [source], which is also the root of such English words as butt and batter.