This expression probably arose from the bat's nocturnal habits and its disorientated flutterings if disturbed by day. The poor eyesight of bats (and less frequently, moles) has been proverbial since the late 16th century.
have bats in the belfry = have bats in your belfry
be eccentric or crazy - informal
This expression refers to the way in which bats in an enclosed space fly about wildly if they are disturbed.
1901 - G. W. Peck - Peck's Red-Headed Boy - They all thought a crazy man with bats in his belfry had got loose.
like a bat out of hell
very fast and wildly – informal
1995 - Patrick McCabe - The Dead School - Like a bat out of hell that Joe Buck gets on out of the apartment and doesn't stop running till he reaches Times Square.
not bat an eye = not bat an eyelid
show no emotional or other reaction - informal
Bat in this sense is perhaps a dialect and US variant of the verb bate meaning lower or let down. The variant not blink an eye is also found.
1997 - James Ryan - Dismantling Mr. Doyle – She did not bat an eyelid when Eve spelled out the unorthodox details of the accommodation they required.
off your own bat
at your own instigation
spontaneously - British
The bat referred to in this phrase is a cricket bat.
1995 - Colin Bateman - Cycle of Violence – She doesn't have me doing anything, Marty. It's All off my own bat.
right off the bat
at the very beginning
straight away - North American
with bated breath
in great suspense
very anxiously or excitedly
Baited, which is sometimes seen, is a misspelling, since bated in this sense is a shortened form of abated, the idea being that your breathing is lessened under the influence of extreme suspense.
an early bath
the sending off of a sports player during a game - British informal
The allusion is to the bath or shower taken by players at the end of a match.
take a bath
suffer a heavy financial loss – informal
1997 - Bookseller - When the yen drops in value, we take a bath. There is no way to change the prices fast enough.
pass on the baton = hand on the baton
hand over a particular duty or responsibility
In athletics, the baton is the short stick or rod passed from one runner to the next in a relay race. The related phrases pick up or take \ up the baton mean accept a duty or responsibility.