For me, the ones I would create are these people:
A new History of Magic teacher and Head of Slytherin House
His name is Jonathan Ewan Clive (First name and middle name from two history teachers of mine and the surname from a Scooby-Doo villain from the Scooby-Doo Show episode "The Spirits of '76" whom I wish wasn't one for a few reasons)
He is the one who gives Hermione her Hogwarts letter and explains everything to her and her parents
Binns who is droning and sleeping during class time and Snape who is a lousy Head of House who encourages the bad behaviour of Slytherins will be at Hogwarts with the former floating around the castle with other ghosts and the latter continuing to be Potions master and later DADA teacher whil continuing to remain to loyal to Dumbledore and spy on Voldemort.
A few good Slytherins
A twin boy and girl in the trio's year who they are friends with, become Prefects in their fifth year instead of Draco Malfoy and Pansy Parkinson
A few new Weasley siblings: Patrick, Victoria and Alexandra - The children order goes like this (Bill, Patrick, Charlie Victoria, Percy, Fred, George, Alexandra, Ron, Ginny)
Characters who were mentioned but never seen that I would add include:
Harry's maternal grandparents the Evanses only at the start of the first book by raising Harry for a while across the Dursleys, but pass away a few years later
Molly Weasley's parents who live at the Burrow
