bitt (n.)

"small piece," c. 1200; related Old English bite "act of biting," and bita "piece bitten off," which probably are the source of the modern words meaning "boring-piece of a drill" (the "biting" part, 1590s), "mouthpiece of a horse's bridle" (mid-14c.), and "a piece (of food) bitten off, morsel" (c. 1000). All from Proto-Germanic *biton (source also of Old Saxon biti, Old Norse bit, Old Frisian bite, Middle Dutch bete, Old High German bizzo "biting," German Bissen "a bite, morsel"), from PIE root *bheid- "to split."

Meaning "small piece, fragment" of anything is from c. 1600. Sense of "short space of time" is 1650s. Theatrical bit part is from 1909. Money sense "small coin" in two bits, etc. is originally from the U.S. South and the West Indies, in reference to silver wedges cut or stamped from Spanish dollars (later Mexican reals); transferred to "eighth of a dollar."

In lexicons of sea language going back to 1759, the bitter end is the part of a cable which is round about the bitts (the two great timbers used to belay cables) when the ship is at anchor (see bitt).

Bitter end of the Cable, the End which is wound about the Bitts. ["The News-Readers Pocket-Book: Or, a Military Dictionary," London, 1759]

So, when a cable is played out to the bitter end, there is no more left to play. The term began to be used c. 1835 in non-nautical use and with probable influence of or merger with bitter (adj.).