Essentially quite a lot
Rowling maintained unprecedented creative control over the Hollywood adaptations of Harry Potter. From her insistence on an all-British cast to the final approval on the script and director, she ensured that the Harry Potter films stayed unerringly faithful to the books
also she retained some control over the HP merchandising including, apparently, refusing permission for a HP-themed toilet seat.
The has continued, somewhat in the Fantastic Beasts arena where Rowling is the screenwriter.
For Rowling, the most important issue was creative control‚ not an easy ask for a first-time screenwriter. Sources say Warners cannot hire someone else to rewrite her script without her approval – a gamble for the studio and a departure from the Potter films, which were written by such seasoned scribes as Steve Kloves and Michael Goldenberg. Rowling also has script approval on subsequent Fantastic Beasts films.
There's even this (unverified) story from 2012 about Theme Parks (Disney & Universal) in Florida... although Disney claims it was about money
Fast forward a couple of years and Disney had somehow made the pitch that if Harry Potter wanted to live on forever, he needed to be in the Disney parks, similar to the kind of popularity that Star Wars and Indiana Jones had been having. The problem is that George Lucas is a fan of Disney and was willing to license his characters relatively inexpensively in exchange for having some creative control and a really cool ride system that no one had ever seen before. JK Rowling had no illusions that she was doing Disney a favor and not the other way around and never significantly reduced her asking price, wanted full creative control, and wanted the people working on the movie to be technical resources to make sure that everything looked exactly right. One of the more contentious issues was around entry into the Harry Potter area, which JK Rowling had insisted should be through the a magical portal at the back of the three broomsticks, something that Disney didn’t think could handle the crowds.
Negotiations went on and off again for nearly three years. Everything came to a head one weekend when JK Rowling was in Florida meeting with the Disney people and had a fairly contentious argument with the Disney staff to the point that she walked out of the meeting. Instead of heading back to the airport, JK decided to instead go over to Universal. She got to the front gate, declared who she was, and more or less demanded to see the president of the park. Within three days, Universal had agreed to just about every demand that she had made, including the price and high level ideas about where it would go in the parkm and what it would contain, and a letter of intent was signed. Designs were immediately started, approved, and construction started within 10 months. What was even more surprising about this is that Universal had just announced the closing of the Back to the Future ride to be replaced by the Simpsons, something that become much more expensive than originally planned as the voice actors had refused to do the ride unless they were paid much more than initially proposed….so Universal was betting quite a bit of their future on the success and popularity of the Simpsons and Harry Potter rides.
The rest, as they say, is history…..except for the bit that is in the future with Harry Potter being so popular in the amusement parks that Universal Studios Japan and Hollywood are lining up with their hat in their hands begging for Harry Potter to come to their park as well.