The meaning of to get off the dime, as native speakers know, is ''to start moving; to stop stalling.'' But what's the origin? And why hasn't the old slang phrase faded along with the value of the 10-cent piece, in an era when hardly anything can be bought for a dime a dozen?
A dime, from the Latin decem, ''ten,'' is the smallest and thinnest U.S. coin. In metaphor, it signifies anything especially tiny. When you are driving, and mean to stop at a precise point, not in a general area -- you stop on a dime.
http://www.nytimes.com/2002/10/06/magazine/the-way-we-live-now-10-6-02-on-language-off-the-dime.html
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 "N***** 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."
https://english.stackexchange.com/questions/281340/when-and-how-did-we-start-getting-off-the-dime