What is the name of Hamlet's father?

Though in the printed script Hamlet's father is listed only as "Ghost", we know his actual name from things that are said in the play. Hamlet's father's name is, in fact, also Hamlet. We know this from a few mentions early on. In Act I, Scene 1, Horatio discusses the history of conflict between Denmark and Norway. In that story, he references "Our last king" and goes on to name him "our valiant Hamlet". At the end of the scene, Horatio makes mention of going to see "young Hamlet" (emphasis added), letting us know that the Hamlet so-named in the play is the son of the old King he was talking about before (and whose ghost they just saw). 

The very next scene begins with Claudius also referencing the name of the departed King, saying "Though yet of Hamlet our dear brother's death the memory be green". Again, he's talking about his brother "Old" Hamlet, now dead, and not his nephew "young" Hamlet, who is the title character of the play. Another way to think of it is that the living Hamlet we see in the play is really "Hamlet, Jr". 

In a parallel to Hamlet and Hamlet (and to make things even more confusing for modern readers), the Norwegian King and prince mentioned in the play are both named Fortinbras. Old Hamlet killed Old Fortinbras years before the play begins. But with the start of the play, Old Hamlet is dead and Young Fortinbras (or Fortinbras, Jr.) sets out with an army to claim some disputed land, which is witnessed and commented on by Young Hamlet.