From trithing (initial /t/ absorbed into preceding north, south, east, west, inner ⟨th⟩ already sometimes /d/ in Middle English), from Middle English trithing, tridinge, from Old English *þriþing, from Old Norse þriðjungr (“third part”), from Proto-Germanic *þridjô (“third”) (English third). Equivalent to third + -ing; compare with farthing (“fourth part”). Cognate with English trithing.
The folk etymology that connects the term to the area a horse-rider could cover in a single day is incorrect, but may have influenced the spelling and pronunciation.