I believe the name originates from the Czech province of Moravia, which in German language is "Mähren". So, the name "Mähr" probably means "the Moravian". To me this appears plausible because my grandfather was born in Thuringia, not too far from Moravia. So, one of his forebears moving into Thuringia from Moravia would have been called "the Moravian" or " Mähr".
(Eingesendet von Walter Mähr)