Famous bearers include the German intellectual Max Weber (1864-1920) and the German physicist Max Planck (1858-1947). This name is also borne by the title character in the Mad Max series of movies, starting 1979.
As a rarer (but older) masculine name it is probably derived from a surname: see Fay 1 or Fay 2.
A diminutive of Baylee, or any name containing the element or sound -bay-.
May also be given in reference to the English word "bay," from the Middle English baye, from the Old English beġ 'berry', as in beġbēam 'berry-tree'.