If the name isn't "son of Morris," it could be Morozov or Morosoff (which actually means frost).
A website I looked at suggested Moryson and Moritsan as possible alternatives, indicating that these surnames are found in Poland and Ukraine, respectively. I don't know whether that's true.
Ellen Morosoff Pemrick
Saratoga County, NY
Researching WEISSMAN/VAYSMAN (Ostropol, Ukraine); MOROZ and ESTRIN/ESTERKIN (Shklov & Bykhov, Belarus); LESSER/LESZEROVITZ, MAIMAN, and BARNETT/BEINHART/BERNHART (Lithuania/Latvia); and ROSENSWEIG/ROSENZWEIG, KIRSCHEN, and SCHWARTZ (Botosani, Romania)