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