"At the Drop of a Dime" Origin

I actually think "at the drop of a dime" resulted from the previously established "at the drop of a hat" combining with the phrase "stop on a dime." This Ngram shows no instances of "drop of a dime" used prior to the introduction of "stop on a dime." According to Etymonline, "phrase stop on a dime attested by 1954 (a dime being the physically smallest unit of U.S. currency)." Additionally, the Ngram demonstrates a clear proportional relationship between the two terms.

American Culture Explained says this about

Drop of a dime:

If you do something at the drop of a dime you do it very fast, pretty much instantaneously, without too much thought, planning, or hesitation. Another way of saying the same thing is “at the drop of a hat”.

The origin of that phrase seem to be the Wild West where dropping a hat was often a sign of an imminent fight. But then again it might also be of Irish origin where they apparently were also ready to fight without much ado.

Here are some examples:
“The situation in Libya is very volatile right now, things can change at the drop of a dime.”

That makes it synonymous with "drop of a hat".
See also: americanidioms

"At the drop of a dime" sure sounds like a mixture of "at the drop of a hat" with "to drop a dime on" (to snitch on). The phrase comes from the old cost of a telephone call (ten cents) by which a criminal could rat out his former friends.