Due to the briefness of the first part and the fact that they were supposed to be brothers anyway, some Hamlet productions have the Ghost and Claudius being played by the same actor. Some see Hamlet's "Look here upon this picture"-speech as proof that this was Shakespeare's intentions and him lampshading it. Note that the guards hang around talking at the end of act 1 scene 1 just long enough for the actor playing The Ghost to jump into his Claudius costume for the start of scene 2.
The Royal National Theatre production with Simon Russell Beale had the actor playing Polonius doubling as the First Gravedigger, accompanied by some Leaning on the Fourth Wall: appropriately enough, given that Polonius was dead, the actor arose from a coffin to play the Gravedigger.
One of the most common misquotations ever; the Alas, Poor Yorick line goes "I knew him, Horatio.", not "I knew him well."
Another is "The Queen doth protest too much" or "Methinks the Queen doth protest too much". The actual line is "The lady doth protest too much, methinks." Also, while most readers think it means a Suspiciously Specific Denial, Gertrude actually means the Lady is making promises she can't keep.
Also, when Hamlet is referenced in media or other fiction, it usually combines his "To be or not to be" with the image of him holding Yorick's skull, which is a different scene altogether. Thematically, this still works as the soliloquy is about death and suicide.
Creator Breakdown: Possibly. Shakespeare had a son named Hamnet who died at age 11. A few years later (we think), Shakespeare decides to adapt a Danish adventure yarn called Amleth, (Anglicized to Hamlet) only he makes the entire thing about death and grief, and changes the happy ending to a tragic one. It's sort of a Role Swap AU of his situation: in the play, a father dies rather than a son, presumably the way he's rather the situation have gone. There's notoriously little evidence of Shakespeare's personal life, and he was at least a partly Absent Father, but still... it's hard to believe there's no link.
Executive Meddling: It is believed that this is why Hamlet had to die in the end; law at the time stated that anyone who killed the king in a work of fiction could not survive, even if the king in question killed his predecessor to usurp the throne.
Word of St. Paul: The text of much of the notoriously garbled First Quarto of Hamlet is widely believed to have been reconstructed from memory by whoever played Marcellus in an early production, because all the scenes with Marcellus in them are closer to their counterparts in later editions, whereas whenever Marcellus is offstage the text starts to read like a best guess by someone who's witnessed several productions but has never bothered to memorise the whole thing. Hamlet's famous soliloquy in this text starts with "To be, or not to be, I, there's the point". The 'mousetrap' scenes in the First Quarto are also very much as they are in later editions, suggesting that the actor doubled up (highly common in Renaissance theatre) and was in those scenes too.
Dawson Casting: MAD joked that Hamlet and Gertrude seem to be the same age. While not quite the case, it is bizarre to have Mel Gibson - 34 at the time of production - play the son of then-43-year-old Glenn Close!
Stunt Casting: Franco Zeffirelli admitted that he cast Mel Gibson as Hamlet in order to draw younger viewers to see the film. Glenn Close was cast likewise because she had recently been in two Box Office successes - Fatal Attraction and Jagged Edge.
During pre-production, it was mentioned that Sean Connery would be playing the ghost of Hamlet's father, who was eventually played by Paul Scofield.
The film was first announced in 1979 - where Richard Gere would play Hamlet, Jean Simmons as Gertrude, EG Marshall as Polonius and Amy Irving as Ophelia. It was cancelled.
1996 film
Acclaimed Flop: Widely agreed to be the best adaptation of Hamlet ever produced but the four hour running time and limited release killed any chance of it being profitable.
Acting for Two: Orlando Seale was Kenneth Branagh's acting double for camera set-ups and rehearsals in addition to having a small role as a boatman. So he had to essentially learn and perform almost the entire play with the other cast members.
Harpo Does Something Funny: When filming the flashback scene during Hamlet's "Yorick" speech, shown in vision only with no sound, Kenneth Branagh's only instruction to Ken Dodd was "Okay, make us laugh", which he did.
Throw It In!: A real blizzard blew in suddenly one day, and they quickly took advantage by shooting the scene where Hamlet's letter is delivered to Horatio - as Michael Farrell could read his lines off the letter.