Spoilers through Harry Potter and the Half-Blood Prince, including its novel.
Severus Snape (Alan Rickman) tells Harry Potter (Daniel Radcliffe) that he is the Half-Blood Prince as he escapes Hogwarts, after Harry has spent a year doing well out of the notes in his old Potions book. Harry learns who wrote them at the moment he least wants to thank him. He has just watched Snape kill Albus Dumbledore (Michael Gambon).
Why Snape chose that name never comes up on screen.
Snape takes the name from his mother
The novel, chapter 30
In the novel, the answer turns up in old copies of the Daily Prophet. Snape’s mother was a witch called Eileen Prince, and his father, Tobias Snape, was a Muggle. Prince is a surname, and half-blood describes his parents.
Snape made himself a title out of his mother’s side of the family. On screen, Harry hears the declaration and nothing else. Every other title in the series is explained in its own film, and this one needed a single extra fact that never arrives.
Harry trusts the Prince’s handwriting more than Snape’s teaching
Harry Potter and the Half-Blood Prince
Slughorn presents Harry with Felix Felicis. Warner Bros. Pictures
Harry’s dislike of Snape does not stop him learning from a version of Snape with no name attached. The notes in the margins get him a perfect Draught of Living Death and a vial of Felix Felicis from Horace Slughorn (Jim Broadbent), and every time the handwriting is right, Harry trusts it more.
Harry finds the risk inside advice that works
Harry Potter and the Half-Blood Prince
Harry stands beside Draco on the bathroom floor. Warner Bros. Pictures
Hermione Granger (Emma Watson) keeps pushing Harry to find out who wrote the notes: “For weeks, you carry around this book, practically sleep with it, and yet you have no desire to find out who the Half-Blood Prince is.” Harry’s defence is “I don’t sleep with it.”
He finds out what else the book can do when he uses Sectumsempra on Draco Malfoy (Tom Felton) without knowing what it does. Draco is left bleeding on a bathroom floor, and Snape has to heal the wounds his own spell made.
One perfect potion was a poor reason to trust every spell in the margins. The book that won Harry a prize nearly led him to kill Draco. Ginny Weasley (Bonnie Wright) takes it off him and hides it in the Room of Requirement.
Snape’s reveal gives Harry a teacher he cannot accept
Harry Potter and the Half-Blood Prince
During the escape, Harry tries Incarcerous on Snape and then the Prince’s own Sectumsempra. Snape blocks the first, knocks Harry flat for the second, and tells him: “You dare use my own spells against me, Potter? Yes, I’m the Half-Blood Prince.”
Harry has spent the year taking advice from the man he hates most.
He valued Snape’s help every time he did not know whose help it was. Losing the family explanation leaves the title sounding grander than the answer Harry gets, which is a name shouted at him in the dark.
Snape named himself after his mother and kept the name in a schoolbook, and Harry meets that side of him without knowing it. The one fact that explains the name stays in the novel, which is a frustrating choice for a film named after it.