Mince
late Middle English: from Old French mincier, based on Latin minutia ‘smallness’.
wiktionary
From Middle English mincen, minsen; partly from Old English minsian, ġeminsian(“to make less, make smaller, diminish”), from Proto-Germanic *minnisōną(“to make less”); partly from Old French mincer, mincier(“to cut into small pieces”), from mince(“slender, slight, puny”), from Frankish *minsto, *minnisto, superlative of *min, *minn(“small, less”), from Proto-Germanic *minniz(“less”); both from Proto-Indo-European *mey-(“small, little”). Cognate with Old Saxon minsōn(“to make less, make smaller”), Gothic 𐌼𐌹𐌽𐌶𐌽𐌰𐌽( minznan, “to become less, diminish”), Swedish minska(“to reduce, lessen”), Gothic 𐌼𐌹𐌽𐍃( mins, “slender, slight”). More at min.
etymonline
mince (v.)
late 14c., mincen, "to chop (meat, herbs, onions, etc.) in little pieces," from Old French mincier "make into small pieces," from Vulgar Latin *minutiare "make small," from Late Latin minutiæ "small bits," from Latin minutus "small" (from PIE root *mei- (2) "small").
From 1540s in reference to speech, "to utter primly or in a half-spoken way as affected delicacy, clip affectedly in imitation of elegance," of words or language, "to restrain in the interest of decorum," 1590s. The meaning "walk with short or precise steps or with affected nicety" is from 1560s. The etymological sense is "to make less, make small." Related: Minced; mincing.
mince (n.)
"minced meat," 1850; see mincemeat. Mince-pie "pie made with minced meat, fruit, etc.," long associated in England with Christmas festivities, is attested from c. 1600; as rhyming slang for eye (n.) it is attested by 1857.
