Ultimately, the name comes from Old English cot, “cottage.” The surname is from the old dative plural cotum, roughly meaning “[he who comes from / dwells at the] cottages.”
What? That’s what the OP asked, isn’t it?
ducks excessively literal-minded head and runs away *
Isn’t it a reference to the Hebrew phrase “He will increase,” which is apparently the meaning ascribed to the name “Joseph”? (Of course, it makes you wonder why Increase Mather’s father didn’t simply name him Joseph. Maybe he didn’t like “Joe.”)
Wiki claims: The stated reason for his first name was “…the never-to-be-forgotten increase, of every sort, wherewith God favoured the country about the time of his nativity.”