k.sherwood 415, I bought the HRP membership when I took two of my grandchildren to London last year, and I was so glad I did. I saved around £40, I think. It really is a good deal. Plus, you can jump the line at places with long lines, like the Tower of London.
Definitely let your grandkid go through the outdoor maze; he will love it! The gardens are always fun to walk through as well. The palace itself is great—my grandkids loved touring it. They should enjoy a lot of it. There's one place where there is an old board game of Fox and Geese set up, and kids can actually sit down and play the game, which my grandkids did. It sounds like you just have one child with you, but one of you could play with him. It's a fun game.
You also might want to stop and have lunch or a snack in the cafe outside. We really enjoyed it there.
The train should be fun for him as well. It's not too long of a ride, and he can see the scenery as the train goes through the countryside. The train will stop in the little town of Hampton Court, where you then walk over the bridge to the palace (about a 5 minute walk). When you're done and want to go back, there is a small cafe in the town of Hampton Court where you could stop for a quick bite if he is hungry.
ETA: you might want to get him a book about Henry VIII before you go. They have a series of books called "Who Was (insert name)?" That are mini-biographies and geared towards younger kids. My grandkids loved them when they were his age, and I know they have one that's called "Who Was Henry VIII?"