Harry Potter and the Goblet of Fire
Originally, Rita Skeeter's role as the nosy investigator in Goblet of Fire was going to be filled by one of Ron's girl cousins. But J.K. Rowling decided (regretfully) to drop her character. Rowling comments, "Rita was always going to be in the book, but I built her up because I needed a kind of conduit for information outside the school. Originally, this girl [and Weasley cousin] fulfilled this purpose." This Weasley cousin was also the famous "phantom plot hole" that J.K. Rowling had to redraft Book 4 to avoid (source).
J.K. Rowling routinely names Goblet of Fire as the hardest in the Harry Potter series to write. She explains, "Goblet of Fire was difficult because by that time I was exhausted. I had been writing as well as being a single mother, as well as trying to hold down a succession of day jobs, I was very tired, Goblet was a bit of a struggle. By the end of that book I really knew I had to take some time off and relax a bit" (source).
While Rowling has stated that she would want to be in Gryffindor House, she also admits that an online Sorting Hat quiz placed her in Hufflepuff. (source).
Mad-Eye Moody's character (or at least, the part with the glass eye that could see through things) was originally going to be a blind Divination instructor named Mopsus. He was "a very early character and he had the power of second sight." Mopsus appeared when J.K. Rowling was drafting Book 1, but she had to cut his character because he was "too good. As the story evolved, if there was somebody who could really do divination […] it greatly diminished the drama of the story" (source).