According to Eric Partridge’s New Dictionary of Slang” it refers primarily to the act of making a phone call—dating from the pre-cellular 1950s, when pay phones required the deposit of ten cents to make a connection.
But why would a fellow want "revenge" just because someone telephoned him?
Originally, dropping a dime on someone simply meant to call them on the phone. But during the late 1950s or early 1960s, a writer of hard-boiled detective stories—Raymond Chandler, James M. Cain, or Mickey Spillane perhaps; no one seems to know who or just when—first used the phrase “drop a dime on” to mean "call the police to inform on a wrong-doer." So now it primarily means to "act as an informer, or to snitch."
Sports announcers have adopted the phrase to mean an “assist” in basketball, derived from the connotation that someone who snitches on a criminal is “assisting” the police.
Someone ought to drop a dime on the Bard of Buffalo Bayou, but it’s probably too late for that to do any good.
A gal who was quite a rip-snorter
Told the guy who had asked to escort her,
“You can have a good time
For only a dime,
But just think what you’d get for a quarter.”