It's derived from the French word dixieme, pronounced roughly "deece-yemm" and meaning one-tenth. That's because a dime is one-tenth of a dollar (ba DING!)
Originally our money system was to be based on tens instead of hundreds, with a dime being as important as a cent or dollar. For ex. a price would be written as "3.8.2", meaning 3 dollars, 8 dimes, and 2 cents. It was a holdover from the old British system that had multiple and sometimes conflicting units. In fact, there were also units called mills (1/10 of a cent), eagles (10 dollars) and unions (100).
However, that cumbersome method was very quickly reduced to cents and dollars, and the dime remains as a kind of oddball coin name.