As for the origin of the last name, Delagrammatikas is definitely a greek one, just like Grammatikas and Grammatikos.
Personally, I can't tell what Dela- means, but 'grammatikas' literally means 'grammar' (from greek 'gramma', 'letter' in english), so maybe such last names may refer to a 'grammarian', also in a more general sense of 'teacher (of letters)'.
Out of curiosity, some very similar last names are found in Italy too (for example Grammatica and Grammatico) and, in the history of Denmark, just think of Saxo Grammaticus (c. 1150 - c. 1220), a famous Danish medieval historian whose work also inspired Shakespeare's Hamlet (for more information, see
http://en.wikipedia.org/wiki/Saxo_Grammaticus): as reported on Wikipedia, Saxo Grammaticus was not his real name, but just an appellation meaning 'Saxon teacher of letters' (literally 'Saxon grammarian').