The word "dime" is an anglicized version of the word "Disme" (pronounced "deem"). meaning tenth. The word is traced back to Simon Stevin Van Brugghe, aka Simon Stevinus, aka Stevin, Stevinus invented the decimal system as a convenient alternative to the fractional system that was being used in mathematics. He published a paper, called "De Thiende" in 1585 under the name Stevin. It was immediately translated to French, and was translated to English by Robert Norton in 1608 and entitled "Disme: The art of Tenths, or, "Decimall arithmeticke".
Although the decimal system won a fair amount of acceptance in the scientific community it did nothing for currency. Stevin's paper was republished in 1634. The article found favor in the American Colonies, and when the United States won its independence, the founding fathers saw the decimal system monetarily as a way to make dramatic change from the past. Among the first coins issued by the United States was the 1792 Half-Disme, and the 1792 Disme. Within a couple of years, the word became "DIME".