There are a ton of Harry Potter video games out right now, movie games, LEGO games, and others. Which games, if any, have had input by J.K. Rowling, and/or had nods of approval?
Guest443 already covered "Book of Spells", so I'm not going to discuss it here.
It appears as though Rowling had quite a bit of creative input on the movie tie-in games. In an interview with The Courier Mail, reported by a fansite (the original article has been taken down), Derek Proud, a producer on the Chamber of Secrets game, had this to say:
"Indeed much of the stuff that we have put in the game which is outside the bounds of the fiction had to be put to [Rowling] so that she could approve it and make sure she was happy with the way we were shaping the Harry Potter universe.
"Rowling seemed to be very happy that we were growing her fictional world and so she gave us a whole lot of extra fictional material which she hadn't used in the books to date.
"For example there is a new creature in the game, called a Gytrash, and it is something she created specifically for us,"
Similar comments were made by Chris Roberts, lead designer on the Order of the Phoenix tie-in game, in an interview with MTV. According to him, Rowling gave them special information, feedback on characterization, and officially canonized some of their inventions (the rules for Gobstones is the example given):
[W]hen the EA team working on the game in London wanted to give players the ability to play wizard games like Gobstones and Exploding Snap, she did more than just say, "OK."
Rowling's books mentioned the games but didn't spell out exactly how Harry and friends play them. Roberts and team dreamed something up. "We wrote the rules up for all these games, sent them off to J. K. Rowling, and she went, 'Yeah, OK, those are the rules,'" he told MTV News during a visit to EA UK's Guildford, England, studio just outside of London. "It's kind of cool. We got to make all the rules."
Roberts is the game designer on the sixth EA "Potter" game, the fifth based on Rowling's books and associated movies (the other game was all about the Harry Potter sport of Quidditch). At this point, Rowling has some confidence in the developers. But they can't get away with everything. "She has written documents for us, explaining how certain things work, why you shouldn't do certain things that we're not supposed to say because they're secret," Roberts said.
And she has put her foot down when need be. Roberts and the team came up with a mission in the game that would have Harry's classmate Neville Longbottom sabotaging a clock tower to get back at Hogwarts professor Dolores Umbridge. "We got this feedback from her saying, 'I really like the mission, but I don't think Neville would do that. I think it would be Dean Thomas. He's much more likely to do it.'" They switched the mission to Dean.
Although I haven't been able to find specific references to other video games based on the movies, it seems likely that Rowling had a similar level of creative input in their development.
Additionally, Rowling confirmed on her website1 that she wrote all of the information for the Famous Wizard Cards that appeared in the Chamber of Secrets video game:
Q: Did you actually write the information that ended up on the Famous Wizard cards?
A: Yes, I wrote the information on the original Famous Wizard cards. As you have noticed, a few of them have now popped up on the 'Wizard of the Month' cards on my website desk.
Regarding the LEGO Harry Poter series, Inside Mac Games interviewed Traveller's Tales Game Director Jonathan Smith in 2011. He was pretty vague, but he did indicate that some agent of Rowling (if not Rowling herself) provided feedback and input on the game (emphasis mine):
Q: What input did you get from JK Rowling and any of the folks involved with the movies?
A: JK Rowling has created the most amazing world, and we are at every moment very much aware of the debt that we owe her. I think that dedication comes through in every detail of LEGO Harry Potter, where the team, as fans, strain every sinew to be true to the rules of the wizarding world and those wonderful characters. In this task, they're extremely well supported by many experts at J K Rowling's agency and Warner Bros., whose feedback and input was nothing less than invaluable.
The interview was explicitly regarding the first game in the series, LEGO Harry Potter: Years 1-4, but it seems likely that this would also hold true for the sequel.
According to the Harry Potter Lexicon, no information in the video games is technically allowed to be called Canon:
I have since learned from [Wizards of the Coast - developers of the Harry Potter trading card game] and from the folks at Electronic Arts [producers of the movie tie-in video games] that companies which create Harry Potter information are forbidden by contract to claim to present "new Harry Potter information." In other words, they can't say that their product is canon, even if it is by our definition.
Whether you consider this information to be canon is really up to you.
1 The page is no longer available on Rowling's site. The provided link is courtesy of the Wayback Machine
The Book of Spells and Book of Potions contain material written by Rowling herself (Source) and so are 100% canon. All the other video games (including the Lego ones) were licensed/published by Warner Bros; that's because technically the games are adaptions of the movies not the books and WB is the one who holds the rights for the HP movie franchise. As such these games had no involvement by Rowling and are to be seen as non-canon or semi-canon depending on your preference. This leads to much annoyance when using the Harry Potter Wikia as they treat everything as if it had the same canon status.
Wonderbook: Book of Spells & Wonderbook: Book of Potions
Practically all of the content in Wonderbook: Book of Spells and Wonderbook: Book of Potions were written directly by J. K. Rowling.
Wonderbook: Book of Spells is an enchanted book that brings spells to life around you, and includes new writing from J.K. Rowling, such as spell descriptions and stories from the wizarding world.
As much as Pottermore (pre-reboot) could be considered a game, it was also largely from Rowling's hand, although the "game" parts of it were less Rowling-made than the rest.
The sorting and wand quizes (if those can be considered games) had the questions and algorithms devised by Rowling.
We had an amazing team working on it, we really did. And I’ve been so involved. In fact, you saw a hint of what’s really fun. You get your wand, you get Sorted into your House, and I think that’s been really popular with users. I devised all of that. I had so much fun with that. I think there are 30,000 and something wand combinations you can get, so you get a really personalized wand.
Devising the definitive questions for the different Houses was a lot of fun. Because there have been so many pale imitations online. It was time for me to do it.
And rest assured, the Sorting Ceremony still uses the authentic algorithm developed by J.K. Rowling.
Other video games
Rowling was somewhat involved in most of the other video games. While none of the other games were directly from her vision in any extant, they all received her approval and she contributed content towards them.
The only parts of the video games that we know for a fact to have come from Rowling is the Famous Wizard Cards. This was first confirmed by Rowling in the FAQ of the old jkrowling.com and then more definitively later at the Warner Bros vs RDR books trial.
Plaintiff: What are wizard cards?
JK Rowling: Within the world of Harry Potter, if you buy a chocolate frog, then you receive a famous wizard card inside the wrapping.
Plaintiff: Did you create wizard cards for use in an Electronic Arts video game?
JK Rowling: Yes, I did.
Plaintiff: And do Warner Bros. and Electronic Arts own the copyright to those cards?
JK Rowling: Yes, I believe so.
Plaintiff: Let's put on the screen Exhibit 34. Can you please tell the Court what Exhibit 34 is.
JK Rowling: This is a list of the famous wizards, well, fictional famous wizards, and their achievements and dates of life and death -- of birth and death, that I provided to Electronic Arts.
Plaintiff: Did you make these wizards up?
JK Rowling: I did. Occasionally there is someone who existed in reality. I've taken some liberties with their biography.
(Warner Bros vs RDR Books, day 1, JK Rowling's direct testimony)
The text from the cards would later appear on Pottermore (pre-reboot).