We don't really know the origin of the surname Tetreau :
Today, the surname Tétreault (Tétrault) is mostly borne in Quebec, Canada. In North America, the patronym took different forms and orthographs even if they all have a common origin. Tatro is a form that we find in United States.
According to Mr. Jean Tosti (French Surname Dictionary) : (my translation of Mr. Tosti's text)
...Today, we find in the Poitou department the surname form Tétreau, Tétrault. The Norman name of Tétrel seems to be the same. Its meaning is uncertain. It could be a diminutive of "têtard" (testart), meaning obstinate (cf. old French "testerie" = caprice). We could also think of a derivative of the verb "téter" to suck, (cf. Occitan word "tetarel" = infant). Also, we could consider a toponym, diminutive of "tertre" mound (hill), with dissimilation of the letter r ...
According to Mr. Gilles Tétreault (ref: Les Tétreau disent..., January 2008, Volume 10, No. 1) : (my translation of Mr. Tetreault's text)
... In our case, the Latin term for "weaving", texere, had moved towards the form of old French teistre, which gave tistre (1). One form of this verb was tétre. The surname Tetrel, which is still found in France, in particular in Normandy, would literally mean, the young weaver, or son of the weaver (2) ...
According to Mr. Gene Tatro (#057) :
...Whatever the origins of the name Tetreau may have been, they seem to be lost in the mists of time... In my own family, it was widely thought to have come from "tête-rouge", or redhead, because many of my family had this characteristic. In fact, Louis Tetreau, our common ancestor, is said to have had red hair...
Until now, the oldest documents found by Josee Tetreault (#61), are burial (1588 and 1589) and marriage (1595) acts at Chatellerault, Vienne, Poitou-Charentes, France.
To conclude, it is complete mystery about the origin of the surname Tetreau.
(1) GODEFROY, Frédéric (1826-1897) - Lexique de l'ancien français
p.509 (1), action de tisser. tissurer, v. a., tisser. tissurier, s. m., tissutier. tisteresce, s. f., femme qui travaille la laine et la toile. tistre, v. a., tisser. tistrer, v. a., tisser. tisture, s. f., tissage il texture. tite, s. m., monnaie de Flandre. *titel, s. m., signe d'abréviation, titulus
(2) JACOB, Roland - Votre nom et son histoire : les noms de familles au Québec