If you would do something quickly and easily, without thinking about it, you would do it at the drop of a hat. Example: "Would you travel around the world if you had the money?" Reply: "At the drop of a hat."

When a hat drops off your head, it falls quickly and suddenly, and that is how fast you would do it at the drop of a hat. Example: "I would take a job like that at the drop of a hat."

You know you would do it, because you have already decided, when you would do it at the drop of a hat. Example: "If you get an extra ticket, let me know. I would go at the drop of a hat."