What is Hamlet's last name in the book Hamlet written by William Shakespeare?

As you say, he's the actual son of Hamlet, but we're not
given any information in the play about what family he belonged to.  Royals usually
didn't go by their surnames in medieval and early modern Europe. It was a real and
deliberate insult to Lous XVI to call him "Citoyen Louis Capet" during his trial and
execution, and the same was true of Charles Stuart, aka Charles I of England. They
thought of themselves as belonging to royal houses more so than tight-knit nuclear
families like everyone else.

I don't think this is what
Shakespeare had in mind when he was writing Hamlet, after all, many of his characters go
by what we would usually associate with first names only: Horatio or Polonius in Hamlet,
for example. Plus, we never learn what King Lear's daughters' last names are either. It
seems to be a detail that Shakespeare wasn't all that concerned
with.