To do something at the drop of a hat means to do it instantly, without any prompting or delay. Someone who starts an argument at the drop of a hat is looking for any excuse to fight. A person who helps at the drop of a hat is unconditionally willing.
In 19th century America and Britain, dropping or waving a hat was a common starting signal for races, fights, and other contests. A referee or official would hold their hat aloft, then drop it — and at that moment the contest began. No countdown, no warning whistle. The action itself was the signal, and it demanded immediate response. The phrase transferred to mean any action taken without delay.
The phrase became widespread in American English from the mid-1800s. It appears in print from the 1860s onwards, typically in contexts involving readiness for conflict or action. Today it is equally used for willing availability — someone who would help at the drop of a hat is reliably, immediately helpful — and for excessive readiness, as in someone who takes offence at the drop of a hat.
No. It can describe positive readiness — 'she would help at the drop of a hat' — or negative impulsiveness — 'he picks fights at the drop of a hat.' The phrase itself is neutral; context determines the tone.
Most languages have equivalent phrases for immediate action. German uses 'auf Anhieb' (on the first strike), French uses 'sur-le-champ' (on the field, meaning instantly). The specific hat imagery is distinctively English.