While I agree with the information provided by J. Diggins regarding the origin of our fine surname, I do have a more plebian explanation.
I had read at one time that Diggins translated to "black head ", similar to dark hair, or black hair.
If you spent all day in a peat bog, you would be pretty dirty, maybe even black with dirt.Ergo, "black head".
Again, this is just my inferrence, and perhaps I am putting too much, and possibly misplaced, emphasis on the "dig" part of our name. However, this has saved me a lot of money over the years, since I never subscribed to any of the direct mail solicitations that claimed "you may be decended from royalty".