Rowling intended to put in a female character named Mafalda (not the Ministry of Magic employee). She was written as being a "black sheep" cousin of the Weasleys, the daughter of the accountant squib relative Ron mentioned to Harry in Harry Potter and the Philosopher's Stone: extremely badly-behaved, unpleasant to be around, nosy and loose-tongued, and to top it off, being sorted into Slytherin. In addition, she would have been the first rival to Hermione, being equally brainy and a show-off, and eavesdropping on the conversations of her fellow Slytherins hoping to relay and impress The Trio. Despite Rowling's fondness for the character, she was ultimately axed due to her age, as Rowling couldn't figure out how to get Mafalda, a first year, to know so much. Rita Skeeter was given many of her qualities as a result.
The Triwizard Tournament was originally intended to have four tasks, each representing the classical four elements of Earth, Air, Fire and Water. Then J.K. Rowling found it too difficult to conceive a task for all four, so Air and Fire were finally combined in the form of the dragons, with the lake representing Water and the maze representing Earth.
Rowling wasn't sure she should include Hagrid's Back Story, since the book was already getting to be so long. She mentioned this to Steve Kloves, who was writing the script for the first movie at the time, and he persuaded her to keep it in, saying, "You can't tell me too much about Hagrid."
Film
Chris Columbus, who had opted out of directing the previous film, had hoped to come back to direct this and rest of the series, but ultimately changed his mind (among other things, his children wanted to move back to America after having been forced to move to England due to his job in the franchise). Still, he has said that he is proud of the work done with all of the films.
At one point, Columbus and writer Kloves advised the producers to adapt the book into a duology of films, but the idea was rejected. Director Newell also decided against it, ultimately cutting some of the book's subplots to make it fit.
Percy Weasley was scripted to appear in the film. Actor Chris Rankin's contract was for only four films, however, and he opted to skip this one in favour of appearing in Harry Potter and the Order of the Phoenix.
Tim Roth was apparently Rowling's first choice for Voldemort, but he was unavailable. John Malkovich came up next, but she shot it down because he wanted a British actor. At this point, wild rumors claimed that Rowan Atkinson was being scouted for the role, to the point his agent had to speak up to deny it.
David Walliams was offered the role of Barty Crouch Jr, but he was busy with Little Britain USA. Another version claims he turned it down because he disliked the character and wanted to play a Hogwarts professor instead.
Rosamund Pike was approached to play Rita Skeeter, but after learning it would be a two-film contract with not much screentime, she declined.
Potter fans campaigned for Henry Cavill to be cast as Cedric Diggory, but Robert Pattinson landed the role instead. Years later, Twilight was set to become a movie, and author Stephenie Meyer expressly said she pictured Henry Cavill when writing Edward Cullen, but he wasn't cast, being considered too old, losing the role to... Robert Pattinson. Again.
Scottish punk band Franz Ferdinand, whose members are huge fans of the series, was approached to play the Weird Sisters along with Jarvis Cocker of Pulp. However, a Canadian folk rock band, Wyrd Sisters, sued Warner Bros, Cocker, and guitarist Jonny Greenwood, so the band in the film ended up nameless. Cocker also wanted making a full album with guests such as Iggy Pop, but he gave up due to the lawsuit.
Concept art was drawn of Winky the Elf, but ultimately she and the entire S.P.E.W. plotline were removed from the film series. As a result, Kreacher's role in the series was significantly cut down, and Dobby would not make another appearance in the movies until his death in Deathly Hallows Part 1.
The lengthy subplot between the Weasley twins and Ludo Bagman was originally included, and there were even reports that Martin Landham had been cast as Bagman. Ultimately, they were cut from the script, with Barty Crouch receiving many of Bagman's leftover lines.
In the film, the first choice for Hermione's Yule Ball outfit was a blue dress like in the book. However, the costume designer found that the blue didn't suit Emma Watson. The dress went through several changes before they ended up with the pink dress instead.
Mike Newell originally wanted the chase with the Hungarian Horntail to travel around the Hogwarts grounds. Newell felt unsatisfied at the results, and mixed with the high budget required, it was instead suggested to have the Horntail chasing Harry around Hogwarts Castle instead.
Warner Bros, Cocker, and guitarist Jonny Greenwood, so the band in the film ended up nameless. Cocker also wanted making a full album with guests such as Iggy Pop, but he gave up due to the lawsuit.