1660s, spinette, "small harpsichord," from French espinette (16c., Modern French épinette), from Italian spinetta, said by Scaliger to be a diminutive of spina "thorn, spine," from Latin spina "thorn" (see spine), so called because the strings were plucked with thorn-like quills [Barnhart]. The other theory (favored by Klein and assigned "greater probability" by OED) dates to early 17c. and claims the word is from the name of the Venetian inventor, Giovanni Spinetti (fl. c. 1503). As "small, upright piano" from 1936.
U.S. underworld slang sense of "$5 bill" is 1925, from Yiddish finif "five," from German fünf (from PIE root *penkwe- "five") and thus unrelated. The same word had been used in England in 1868 to mean "five pound note" (earlier finnip, 1839).
c. 1400 (late 13c. as a surname), from Anglo-French spinache, Old French espinache (14c., Modern French épinard, from a form with a different suffix), from Old Provençal espinarc, which perhaps is via Catalan espinac, from Andalusian Arabic isbinakh, from Arabic isbanakh, from Persian aspanakh "spinach." But OED is not convinced the Middle Eastern words are native, and based on the plethora of Romanic forms pronounces the origin "doubtful."
Popeye, the spinach-eating superman, debuted in 1929. Old folk etymology connected the word with Latin spina (see spine) or with Medieval Latin Hispanicum olus. For pronunciation, see cabbage. In 1930s colloquial American English, it had a sense of "nonsense, rubbish," based on a famous New Yorker cartoon of Dec. 8, 1928. Related: spinaceous.