mite (n.1)

"tiny animal, minute arachnid," Old English mite, from Proto-Germanic *miton (source also of Middle Dutch mite, Dutch mijt, Old High German miza, Danish mide) originally meaning perhaps "the cutter," in reference to its bite, from Proto-Germanic *mait- (source also of Gothic maitan, Old High German meizen "to cut"), from PIE root *mai- (1) "to cut" (see maim). Or else its original sense is "something small," and it is from PIE root *mei- (2) "small," in reference to size.

mite (n.2)

"little bit," mid-14c., from Middle Dutch or Middle Low German mite "tiny animal," from Proto-Germanic *miton-, from PIE root *mei- (2) "small," and thus probably identical with mite (n.1). Also the name of a medieval Flemish copper coin of very small value, proverbial in English for "a very small unit of money," hence used since Wyclif to translate Latin minutum from Vulgate in Mark xii.43, itself a translation of Greek lepton. French mite (14c.) is a loan-word from Dutch.