The idiom 'At the drop of a hat means':

To do something willingly and instantly

The correct answer is Option (1) → To do something willingly and instantly

The idiom "at the drop of a hat" means to do something immediately or without any hesitation. It implies a willingness to act promptly without needing much, if any, notice or preparation. The phrase is often used to describe someone who is ready to respond instantly to a situation or request.