As a medieval given name this is a Norman form of the French name Thierry, from Germanic Theodoric, from þeud ‘people, race’ + rīc ‘power, ruler’. This was adopted by the Normans and introduced by them to Britain. In modern English use it seems at first to have been a transferred use of the surname derived from the medieval given name, and later to have been taken as a pet form of Terence.
Probably an Americanized spelling of German Kötner or Köthner, status names for a cotter, derived from Middle High German, Middle Low German kote ‘shelter’, ‘cottage’, specifically of a kind that was rented on a daily basis and had little or no land with it. See Koth 1.