fin (n.) Look up fin at Dictionary.com
Old English finn, from Proto-Germanic *finno (cognates: Middle Low German vinne, Dutch vin), perhaps from Latin pinna "feather, wing" (see pin (n.)); or, less likely, from Latin spina "thorn, spine" (see spine).

U.S. underworld slang sense of "$5 bill" is 1925, from Yiddish finif "five," from German fünf (see five) and thus unrelated. The same word had been used in England 1868 to mean "five pound note" (earlier finnip, 1839).