chosen 1786 as name for U.S. 10 cent coin, from dime "a tenth, tithe" (late 14c.), from Old French disme (Modern French dîme) "a tenth part," from Latin decima (pars) "tenth (part)," from decem "ten" (see ten).
The verb meaning "to inform" (on someone) is 1960s, from the then-cost of a pay phone call. A dime a dozen "almost worthless" first recorded 1930. Phrase stop on a dime attested by 1954 (a dime being the physically smallest unit of U.S. currency).
[verb sense from the dime dropped into the pay telephone for the call to the police]