According to my Nissen relatives in Thisted, Denmark Nissen means son of Nis.
My father emigrated to Britain in the 1960s. All the Nissens I'm related to are farmers or cabinet makers from Jutland, and have been for generations and speak a Danish dialect. Its def. a non aristocratic name. I can remember visiting my great uncle Lars who ran a farm, single handed, until well into his 80s.
As the German Danish border has changed a great deal over the last 150 yrs (Denmark owned Hamburg at one point) some Nissens are in Germany. In fact I believe that one sits on the commitee trying to make Hamburg Danish again. This caused problems during the 2nd World War. Some Nissens, and other Danes who lived in Hamburg, who had Danish sympathies (Denmark was occupied), were either forced to sign up or sent to camps as they refused to fight for Nazi Germany.
There is a make of snaps in Germany called 'Nissen'.
Not very specific info but it might be of interest.