Enforced Method Acting: The Hogwarts students had three weeks of dance lessons to prepare for the Yule Ball scenes. Daniel Radcliffe however only had three days, which worked in his favour - since Harry is supposed to be an awkward dancer. This is also the reason Harry is only filmed from the waist up during the dancing, to hide his fumbling feet.
Fake Nationality: Marginal example. Clemence Posey is French but speaks English with a British accent, and so adopted a French accent whenever Fleur speaks English. Likewise Frances de la Tour has French ancestry but was born and raised in Britain - so she too put on a French accent to play Madam Maxime.
Fandom Nod: The pronunciation of "Hermione", which was finally clarified by Hermione herself in this book — then the films were released, so everyone knew how to pronounce it anyway.
J.K. Rowling intended to put in a female character named Mafalda (not the Ministry of Magic employee). She was written at being a "black sheep" cousin of the Weasleys'; 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 into 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 limiting her story potential. Mafalda would be the daughter of the accountant relative Ron mentioned to Harry in Harry Potter and the Philosopher's Stone. 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.
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 (the one the fandom had a collective fit over).
The book was nearly split into two separate films, but Mike Newell decided against it - cutting some of the book's subplots to make it fit. The seventh book was eventually split into two films.
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.
Rosamund Pike was approached to play Rita Skeeter but she declined. Eventually Miranda Richardson was cast.
The lengthy subplot between the Weasley twins and Ludo Bagman was originally included. There were even reports that Martin Landham had been cast as Bagman. Ultimately they were cut from the script, and Barty Crouch has many of Bagman's leftover lines.
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.
Write What You Know: At the height of the Religious Controversy for Harry Potter, Goblet of Fire in particular was heavily criticized for the religious imagery of Voldemort's resurrection. Considering Rowling herself is actually a Christian, the imagery was definitely intended.