Rufus as a given name always had the meaning "red haired" or maybe "red beard", and also could have derived from families with "ruddy" complexions (where the blood vessels made the skin look reddish or very pink in color).
In my family we always thought the name was Anglo-Saxon or Scandinavian in origin. Let's say the person's first name was Tom or Will. When surnames became common in Europe after the year 1250, someone already known as "Tom the Red" or "Will the Ruddy" might have become Tom Red, Tom Reddy, and Will Ruddy. I think the same thing happened with the French word "Roux" or Italian "Ruffino". RUFUS is jsut Latin for "red".
Is this the same idea with the use of RUFUS as a surname ?