If you want to get really technical, the Danish Royal Family surname is Schleswig-Holstein-Sonderburg-Glücksburg. But back in Hamlet's day, the time of medieval Kings, there was no house name or surname.
(As you may know, the surname of British Royalty used to be Saxe-Coburg-Gotha, but, because of World War I, it was changed to Windsor in a rejection of German associations in 1917.)
So, Hamlet's father was King Hamlet, and his son, also named Hamlet, was Prince Hamlet or Hamlet, Prince of Denmark.
The same probably goes for everyone in the play: Claudius is just Claudius, Gertrude, Gertrude, etc. At best, just to differentiate individuals with the same name, people may have been named something like this: if your name was John and you were the son of Eric, you would be John Ericson.
Hmmm, Hamlet Hamletson? Doesn't sound right somehow. Let's leave it as Hamlet.