According to The Word Detective get off the dime is actually older in origin than "stop on a dime", it dates back to the 20's and probably originated during the days of dance halls and "taxi dancer":
- “Get off the dime” has been around since at least the 1920s, and today it’s generally used to mean, as defined by the Historical Dictionary of American Slang, “to take action after a period of indecision or procrastination; to act” (“Congress [should] get off the dime and adopt the … budget proposal before it,” President Ronald Reagan, 1982).
- Since a dime is a small unit of money and fairly easily to come by, this small coin has played a much larger role in US slang than, for instance, the hundred-dollar bill.
- To “drop a dime on someone,” for instance, means to inform on them, usually by tipping off the police, and originated back in the 1960s when a call from a public telephone cost ten cents. “Dime” has also found a home in the slang of drug users, where a “dime” or “dime bag” has long meant ten dollars worth of a drug. The small size of a dime has also been used as a metaphor for “a small spot,” as in “stop on a dime” or “turn on a dime” when speaking of motor vehicles (or politicians).
- “Get off the dime” dates back to the days of dance halls and “taxi dancers,” women employed by the halls to dance with strangers, usually for ten cents per dance (a grim occupation immortalized in the 1930 Rodgers and Hart song “Ten Cents a Dance”).
- A contemporary account, published in 1925, explains the phrase: “Sometimes a … [dancing] couple would … scarcely move from one spot. Then the floor manager would cry ‘Git off dat dime!'” Similarly, “dancing on the dime” meant to dance very closely with very little movement, behavior that might well attract the attention of the Vice Squad and get the hall closed. Thus “get off the dime” referred both to the the customer as the “dime” he had paid and to the small spot (“dime”) on the floor where the couple seemed frozen.
Atually according to The Dictionary of American Slang, Fourth Edition ByHarperCollins get off the dime is:
- (1925+); an alteration of the expression stop on a dime, used to praise the brakes of a car.
Ngram shows usage of the two expressions are from the late 20's/early 30's.
The following extract suggests that they may have come into usage about the same period and in the same context:
- Thanks to Jonathan Lighter's "Historical Dictionary of American Slang," we have the activity that coined the phrase. Carl Van Vechter, one of the earliest modern dance critics and author of the 1926 novel "Nigger Heaven" - a title nobody would use today - described the scene in a taxi-dance hall: "Sometimes a … couple would scarcely move from one spot. Then the floor manager would cry, Git off dat dime!"
- To dance on a dime was to grind bodies tightly together in clothed but sexual contact, without moving from that spot; taxi dancers working for a dime (immortalized in the 1930 Lorenz Hart lyric "Ten Cents a Dance") were exhorted by their bosses to keep the customers moving. Thus, to get off the dime came to mean "to get moving." (I get a kick out of finding that out. What a rich language we speak, often without knowing its resonances.)
(ytlcommunity.com)