Bring on the tough stuff - there’s not just one right answer.
The big question of all of English literature: why does Hamlet drag his feet so long to avenge his father's murder? Sure, plenty of scholar-types have tried to answer that, but maybe they've missed something. Give it a shot.
What is the role of theater within Hamlet? What is the purpose of the Hecuba speech, the play-within-the-play, and Hamlet's advice to actors? What practical purposes do theatrical moments serve in the plot? What symbolic purposes do they serve? Does theater really "hold, as twere, a mirror up to nature" (3.2.2)?
Hamlet is full of madness, both real and feigned (maybe). What's the diff between the mad and the sane characters in the play, especially in what they say and how they say it? What are the similarities and differences between Hamlet's madness and Ophelia's?
Hamlet's conflicts take place in the context of a single family's domestic problems, and also in the context of political decisions that affect an entire country. How do the familial and political levels of Hamlet interact? Where do they reinforce each other, and where, if ever, do they contradict each other?
Almost from his opening lines, Hamlet is obsessed with suicide. He never does it, but he sure thinks about it a lot. How do Hamlet's reasons for avoiding suicide —and his attitude towards his own death —change throughout the play?
One of the more famous lines in Hamlet is, "To thine own self be true, and it must follow, as the night the day, thou canst not then be false to any man" (1.3.1). Which of the characters in Hamlet are true to themselves? Does that prevent them from being false to one another? Does the meaning of this quote change for you when you consider that it's spoken by Polonius, one of the play's most frequently mocked characters?
Hamlet seems pretty grossed out by his mother's sexuality. Why does he dwell so obsessively on the "incestuous sheets" of her marriage bed? What other weird sex-obsessions does Hamlet have?
It seems like half the characters in Hamlet are foils for Hamlet, and the rest of them are foils for each other. How do the different foils bring out different aspects of Hamlet's character? What other effects does all this doubling produce?
Is Hamlet's reaction to his mother's remarriage reasonable, or are his standards of fidelity too high? In the play-within-the-play, which Hamlet himself revised, the player Queen vows never to marry again: "Such love would need be treason in my breast" (3.2.2). Does Hamlet expect his mother to remain single for the rest of her life? If so, why?
As he's dying, Hamlet begs Horatio to tell his story. Do you think the story Horatio will tell is the same one the readers or the audience have just experienced? Is Horatio capable of telling Hamlet's true story?
At the close of the play, Fortinbras says, "Bear Hamlet like a soldier to the stage, / For he was likely, had he been put on, / To have proved most royal" (5.2.4). Fortinbras, who never met Hamlet, characterizes him as a good soldier who would have made a good king. Do you think Hamlet would have made a good king? Why or why not?