An Etymological Dictionary of the German Language/Annotated/Bühne
Bühne, feminine, ‘stage, gallery, orchestra,’ from Middle High German büne, bün, feminine, ‘ceiling of a room (a meaning still preserved in Swiss), board, lath’; the latter is at all events the primary meaning. Perhaps Anglo-Saxon binn, ‘crib, box,’ English bin, are allied by gradation to Middle High German büne. The origin of the words has not yet been explained.