There are a number of theories about the origin of the surname but it seems the most prevalent idea is that this name is derived from French Huguenots who fled France to England and thence to North America and elsewhere.
"Dy" seems to be an abbreviated form of "de Haye" and "de Haye" would indeed be pronounced in French as "die," long "i," silent "H.""Ment" which is also spelled "mond" by some families, is a suffix in French which compares to "ly" in English.So, if the name were to be translated from French to English it would mean approx. "of Haye."Haye is a town in Normandy, France.
My Dyment ancestors came from Devonshire England to PEI, Canada and it looks like they were involved in Protestantism, particularly, Congregationalism, a denomination begun by Calvinist (French) Protestants in Britain so I am inclined to believe they were indeed Huguenots from Haye, France who first arrive in Britain in the 1500s.