What does 'at the drop of a hat' mean?
Generall, "At the drop of a hat" means that someone/something can react or happen as quickly as someone dropping their hat. ie: Jessica can start crying at the drop of a hat.
This term originated in sports - before they invented starting guns, the race or event would start when the official held their hat in an outstretched arm and suddenly lowered the arm straight down. That was called "dropping" the hat.