Thank you all! We've been to London several times, but our kids were 11 and 14 the first time we went and they enjoyed the British Museum and even the British Library. He will still be 9, which I know is young, so this will be a different approach. That being said, he's flown cross country to visit family, is an adventurous eater, and has lots of stamina, so I think we'll be fine. Going to let him call the shots about how much he's able to do and take time for frequent breaks.
Definitely going to hit Wagamama's for ramen (his favorite thing), and we'll be sure to try The Golden Union, as Hamley's will be fun for souvenir shopping. He has to have fish and chips! I'm sure Nando's, Pret, and Bill's will go over well too.
As for Hampton, my husband and I were in the Loire Valley in September and I sent him pics of a couple of the chateaux. One had a maze, so he decided he wants to see a castle with a maze as they fascinate him. I think at this age the Buckingham and Windsor tours would be too dry for him. Hampton (which I've never seen either) has a kids' audioguide tour, the maze, and a huge adventure playground. Food on the premises, and families are welcome to picnic. Plus he's never been on a train so he'll enjoy doing that to get there. (He'll love helping decipher the Tube maps too.) I'm going to feed him some age-appropriate history before we go so he'll understand the historic sites better.
Will look into Premier Inns and the Resident, as I think a "standard" hotel in a good location is our best bet. And I'm already researching playgrounds (even found one on the river near the Tower). Kensington will be a definite. Also have the Eye, the Tower Bridge Experience, and the horse guards on the list. And yes, Stamford Bridge. He and dad watch Chelsea every week.
Again, thanks for all the recommendations. As I said, been to the UK a number of times, but not with a kid this young.