From Ancestry.com: “North American form of French Bresset, from a pet form of the personal name Brès, a variant of Brice,” in other words ‘Little Brice.’ Maybe from the 5th Century French Bishop of Tours, Saint Brice.
The Bressettes were early settlers in French Canada. The earliest ancestor I have found was Jean Baptiste Brissett (1793-?). Generations of the family moved back and forth between Quebec and New York State. Most worked in textile mills.
The name has been found variously spelled. In fact, my father-in-law said he had to legally correct his original birth certificate because of a misspelling. I have found the name spelled (or transcribed in indexes) the following ways: Bressette, Bresette, Bresett, Brissette, Brisset, Besset & Briessette