It should be a completely new story with a new cast of characters, and should take place after Harry’s kids are already out of Hogwarts. Harry, Hermione and Ron should only be tangential characters working in the ministry, like Cornelius Fudge or Rufus Scrimgeour were in the books and movies. There could still be a number of familiar characters, like McGonagall or Neville, who are working at Hogwarts, but for the most part it should be a fresh start.
I don’t like the idea of a prequel show. We already know all the big events of the 20th century (Grindelwald, Voldemort, Voldemort 2.0), we know which important characters die (in some cases we even know exactly when and how they die), we know who marries whom, and which side wins. A series about Harry’s parents during their time at Hogwarts for example wouldn’t be very exciting for those reasons. Voldemort as a character isn’t interesting enough for his own show, and Grindelwald’s story is already being told through the Fantastic Beasts movies. They could make a series taking place in earlier centuries I guess, but one of the biggest appeals of HP has always been about this whimsical magical society hidden from our modern world, and I feel a lot of that charm would be lost if the story took place in the middle ages or something.
The only other thing they could do that would feel both familiar, but also fresh and exciting, would be an AU retelling of the HP series, where Harry gets sorted into Slytherin, for example, instead of Gryffindor. There are lot’s of fanfics with that premise, and I think a tv show based on that idea could be pretty amazing. It would feel very familiar for fans, and would involve all those beloved characters from the books, but – unlike with a prequel or another adaptation of the books – the audience would have no idea where the story goes, or how it ends. Would Harry befriend Draco Malfoy if he got sorted into Slytherin, for example? Would he still try to save the Philosopher’s Stone? How would Snape react? Does Sirius still escape in Harry’s third year? There are lot’s of exciting ways the story could go.
Unfortunately, if the Game of Thrones prequel is anything to go by, this series (if it actually happens) will probably be a prequel too, taking place during Voldemort’s rise in the 70s, when Harry’s parents went to Hogwarts. That probably sounds like the obvious solution for HBO.