The Trauring surname originated in the late 19th century in the Galicia region of the former Austro-Hungarian Empire, currently part of Poland.
In the early years of the 19th century in Galicia, Jews were required for the first time to take on surnames. This was in order to make it easier to tax them, as well as conscript them into the army. Sometime the people were allowed to choose their surnames, although always with the approval of a local bureaucrat. Sometimes these bureaucrats would request bribes in order to give better names, and sometimes they would maliciously hand out bad names. Therein probably lies the origin of the original surname Traurig, which in German means sad. It's unlikely a family would have chosen the name Traurig on their own.
In any case, as members of the Traurig family in Kanczuga moved to other cities in the region (Rzeszow and Lancut for example), they would change their surnames, to Trauring - which means 'wedding ring' in German. A bit more palatable.
Just to be clear, there are many Traurigs out there who are not related to the Traurings, as it might have been a common name to give to a family that could not pay a bribe, but all Traurings originated as Traurigs in the 19th century.