chosen 1786 as name for U.S. 10-cent coin (originally of silver), from dime "a tenth, tithe" (late 14c.), from Old French disme (Modern French dîme) "a tenth part" and directly from Medieval Latin decima, from Latin decima (pars) "tenth (part)," from decem "ten" (from PIE root *dekm- "ten").
The verb meaning "to inform" (on someone) is from the 1960s, from the then-cost of a pay-phone call. Alliterative phrase a dime a dozen "almost worthless" is recorded by 1930 (as an actual price, for eggs, etc., by 1861). Phrase stop on a dime attested by 1927 (a dime being the physically smallest unit of U.S. currency); turn on a dime is by 1913. Dime-store "retail outlet selling everything for (more or less) 10 cents" is by 1928.