Based on what Rowling has said in interviews, it would seem that the original idea was created by Jack Thorne, John Tiffany, Sonia Friedman, and Colin Callender before Rowling was even involved. Like by most of the other licensed works Rowling was on the creative team and had veto power, but her main contribution seems to be sharing her pre-existing notes on the next-generation with the creative team.
Rowling originally had not wanted to make a stage adaptation. Producers Sonia Friedman and Colin Callender came up with the idea for The Cursed Child and successfully pitched it to Rowling.
Over the years I have received countless approaches about turning Harry Potter into a theatrical production, but Sonia and Colin’s vision was the only one that really made sense to me, and which had the sensitivity, intensity and intimacy I thought appropriate for bringing Harry's story to the stage. After a year in gestation it is exciting to see this project moving on to the next phase. I’d like to thank Warner Bros. for their continuing support in this project.
(J.K. Rowling Announcement - December 20, 2013)
You also announced that you're going to collaborate on a theatre production.
J.K. Rowling: Yes that was a really interesting idea that Sonia Friedman came up with. I've been so resistant for a long time about theatre productions. Quite a few people wanted to do a Harry Potter musical. I didn't really see Harry as a musical so we said no to all of that, but Sonia came along with a very thoughtful, very interesting idea. I'm quite excited about that.
(J.K. Rowling: Author and Philanthropist. Wonderland Magazine, February/March 2014, p 184.)
Why a play? Was it J. K. Rowling’s idea?
Sonia Friedman: It was absolutely Colin’s and my idea. We knew that many other producers had approached her and she had rejected their pitches. But that’s because they were all ideas about musicals or arena spectacles. We went to her with the simple idea of a straight play. We were clear we didn’t want to adapt a novel, and we suggested exploring how Harry, an orphan, would cope as an adult and a parent.
Colin Callender: We went up to Edinburgh four years ago and sat in a boardroom and talked about fathers and parenting for a while. We said we felt she had created a fully dimensional world, and there were things about the characters she hadn’t revealed. We didn’t hear anything for a bit, then got the call to say, “Let’s go to the next stage.”
(NY Times - Why J. K. Rowling Endorsed ‘Harry Potter and the Cursed Child’ for the Stage)
She [Sonia Friedman] had been in tears when she knocked on J K Rowling’s door in Edinburgh, readying to make the big pitch; her father had lived in the neighbouring street. “I’d been thinking, ‘If only my dad were alive now, perhaps he’d finally be proud of me’. So, I talked a lot about him when we first met. We didn’t talk about merchandise and box-office. I talked about what I wanted an audience to feel. I said, ‘I think this needs to be about a dad who doesn’t know how to be a dad’.”
The rest is theatrical history, still in the making. So, what next? The show will get a Broadway incarnation next year but Friedman says she won’t be “settled until we have 10 productions around the world”. Will it be like a blockbuster musical? “Yes, but it’s a play, and seven hours in the theatre, so we have to get it right.”
(The Telegraph - Cursed Child mastermind Sonia Friedman: 'I was in tears when I met JK Rowling')
Speaking at the opening gala of Harry Potter And The Cursed Child in the West End, the author said finally agreeing to go ahead with a stage show had been a "daunting" prospect.
She said: "I think I've had probably three offers a week for the past decade to do either a musical or a play or an ice show or an opera - you name it, I've been asked to do it."
She said meeting producer Sonia Friedman - whose credits include The Book Of Mormon and Funny Girl - convinced her that "this is the one".
(Sky News - Rowling Rejected Hundreds Of Potter Spin-Offs)
Sonia Friedman: She had no intention of doing a theater show, but we when explained what we wanted to do, and how we were going to do it, she seemed to be fine.
Colin Calindar: You know, we didn't talk story, we didn't talk about plot.
(Harry Potter and the Cursed Child Premiere - Rebecca Williams reports)
John Tiffany: I was approached by Sonia and Colin, and they told me that they'd been to meet with Jo, and Jo had given them the okay to move forward with the idea of putting Harry onstage and exploring the adult Harry, and specifically looking at what happened to somebody who'd had the childhood that Harry had, and how he then became a parent, a father himself.
(Extended transcript: J.K. Rowling and the creative team behind "Cursed Child")
It was Ms. Friedman and Mr. Callender who, six years ago, brought the idea of a play to Ms. Rowling, even though she had consistently rebuffed proposals to create stage versions of her novels. “Most of the ideas were about musicals, which I don’t love,” Ms. Rowling said, “or redoing the books on stage. I wasn’t interested in doing Harry in every medium.”
Their proposition was different. They suggested extending the story and creating a new work, which intrigued Ms. Rowling.
“We talked about loss, fear, bereavement, what it’s like to try to make a family when your own is poor or nonexistent,” she said. “I was really interested in making something more reflective than had been possible in the films. I don’t think we ever deviated from those themes.”
(NY Times - How Much Magic Can ‘Harry Potter and the Cursed Child’ Make on Broadway?)
For ten years I said no to proposals to adapt Harry for the stage, usually as a musical, and using the existing books. So when I met producers Sonia Friedman and Colin Callender, I wasn’t sure what I was going to hear. I only knew that they wanted to do something new, which was intriguing, because I had no desire to go endlessly back over old ground.
("Answers to Questions", jkrowling.com, may 30 2018)
The general idea for the play was then thought out by director Jack Tiffany and writer John Thorne with Rowling's help. Thorne did all the writing, with Rowling merely approving it.
What changed your mind?
J.K. Rowling: Meeting Sonya Friedman, Jack Thorne and John Tiffany changed my mind.
I knew there was other material there, I gave it all to them and they've turned out the play.
(BBC News - JK Rowling: I did say 'never say never' for Harry Potter)
His [Jack Thorne's] involvement in Potter came through Let the Right One In’s director, John Tiffany, and the producer Sonia Friedman, who recommended Thorne to J K Rowling. Tiffany and Thorne trooped off to see Rowling and find what new material they could develop for Thorne to script. They came up with the idea of a sequel tracking the adventures of Harry’s son Albus and his best friend, Scorpius Malfoy.
Thorne had long been a Potter addict and his anxiety about treading on such hallowed ground was assuaged by Rowling’s involvement. “I had a big advantage – my first reader was John, and my second was Jo. If you’ve got the person out of whose head these characters came, then you go, ‘OK, I can make some choices. If they’re the wrong ones, she’ll say.’ And she did. I’m pretty sure I could have been fired at any time.”
(NewStatesman - From Harry Potter to Jimmy Savile: Jack Thorne on the darkness that defines his dramas)
I would love to be with you tonight to accept this award alongside my co-collaborators, but something else was happening in the Wizarding World on the other side of the Atlantic. I am incredibly proud of Harry Potter and the Cursed Child, but Jack Thorne and John Tiffany must take all credit for shaping and crafting it from the very very first story into something very special. My endless thanks to them.
(Rowling's statement at the 2016 ES Theatre Awards)
I told John and Jack what I thought had happened to Harry, Ron and Hermione in later years, explained how focused I was on Harry’s son Albus, who’d been given the burden of not one, but three legendary names, and together we created the story that Jack wrote.
("Answers to Questions", jkrowling.com, may 30 2018)
Rowling did not do any of the writing herself
She was clear from the beginning that she was not a playwright and wouldn’t write it, and that she would only do it if we found a playwright she approved.
(NY Times - Why J. K. Rowling Endorsed ‘Harry Potter and the Cursed Child’ for the Stage)
Jack Thorne: I'm stuck - and this is the point when I normally e-mail you!
J.K. Rowling: And I say 'oh God, I had that problem in Goblet', fail to offer any decent solution and you do it!
(Twitter - @jk_rowling)
Just to say that this a show actually written by an awful lot of people. A lot of people who you have seen come up tonight, a lot you haven’t. I feel very, very privileged to be apart of it. Thank you Sonia, thank you Collin, thank you, thank you, John Tiffany.
(Jack Thorne - 2017 Oliver Awards)
The ideas for the books came from you. The ideas for the movies came from the books. This was extending the story into the future. Who did that? Which of the three of you was most responsible for the storyline as it weaves its way into the future?
J.K. Rowling: The developing the story, I think, was very collaborative between the three of us. I, for obvious reasons, had power of veto over everything. I could say, "No, that didn't happen." But no, it was the three of us. But the play is Jack's play. Jack did the writing. Jack did the heavy lifting. And he did it beautifully. And I couldn't be happier.
(Extended transcript: J.K. Rowling and the creative team behind "Cursed Child")
What in Mr. Thorne’s writing had surprised her? Ms. Rowling pointed to the way he had imagined the character of Scorpius Malfoy.
“He is such a beautiful character, and in many ways the emotional heart of the play,” she said. “And such an amazing foil for Albus, who is tortured and self-involved.”
(NY Times - How Much Magic Can ‘Harry Potter and the Cursed Child’ Make on Broadway?)