half-dime (n.)

Old English half, halb (Mercian), healf (W. Saxon) "side, part," not necessarily of equal division (original sense preserved in behalf), from Proto-Germanic *halba- "something divided" (source also of Old Saxon halba, Old Norse halfr, Old Frisian, Middle Dutch half, German halb, Gothic halbs "half"), a word of no certain etymology. Perhaps from PIE root *skel- (1) "to cut," or perhaps a substratum word. Noun, adjective, and adverb all were in Old English.

Used also in Old English phrases, as in modern German, to mean "one half unit less than," for example þridda healf "two and a half," literally "half third." The construction in two and a half, etc., is first recorded c. 1200. Of time, in half past ten, etc., first attested 1750; in Scottish, the half often is prefixed to the following hour (as in German, halb elf = "ten thirty").

To go off half-cocked in the figurative sense "speak or act too hastily" (1833) is in allusion to firearms going off prematurely; half-cocked in a literal sense "with the cock lifted to the first catch, at which position the trigger does not act" is recorded by 1750. In 1770 it was noted as a synonym for "drunk." Bartlett ("Dictionary of Americanisms," 1848) writes that it was "a metaphorical expression borrowed from the language of sportsmen, and is applied to a person who attempts a thing in a hurry without due preparation, and consequently fails."

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.