From Middle English mud, mudde, mode, probably a borrowing from Middle Dutch mod, modde or Middle Low German mudde, ultimately from Proto-Germanic *mud-, *mudra-, perhaps from Proto-Indo-European *mū-, *mew-. Cognate with Saterland Frisian Mudde, Middle High German mot, Swedish modd. Compare also suffixed variants West Frisian modder, Dutch modder, German Low German Mudder, German Moder.
Alternative etymology suggests the Proto-Germanic word is possibly borrowed from a language (compare e.g. Finnish muta, Northern Sami mođđi, from Proto-Uralic *muďa).