origins
Bunce Genealogy and History (link below) reports that this surname arose among male-line descendants of Pons FitzWilliam.
Perhaps a respelling of Bunts.
Probably an altered spelling of Swiss German Bunz or Bünz, from Alemannic bunz "little barrel", hence a nickname for a short fat man, or of German Banz, or from pet form of an Old High German personal name Bun(n)o, of unexplained etymology.