Hampton Court Palace or Windsor Castle

I love both Windsor castle and the town of Windsor. It is also just a short taxi ride to Heathrow airport, so you could stay the night.

I had a similar opportunity a few years ago and saw the Tower of London, cruised the Thames, walked around Westminster and visited the British Museum. If you haven't seen the major sites of London, a day there might be a better choice. I'm not sure but I think you'll end up going through London if you use anything but a taxi .

They are equally interesting, so I'd suggest Windsor Castle due to it being so close to Heathrow. If you can be there at 11 AM (Mon-Sat from Apr-July) you get the added attraction of the 'changing of the guard.'

We visit Hampton Court every time we are in London. There is so much to see there, especially when you can also wander the gardens. We do Windsor about every second visit- still interesting. Depends whether you are more interested in wandering around the interior of a (very large) old castle/manor house (Hampton), or more the exterior with soldiers on guard duty, the chapel and a cute town (Windsor). Hampton Court is more set up for the tourist with restaurants, gift shops etc in the facility; whereas at Windsor you look for that outside the castle

I would go to Windsor it's a 15 minute taxi ride. You won't have trouble finding places to eat at, there are plenty of places. There is a nice pub as you exit the castle thru the Henry VIII gates the pub is across the street...The Horse and Groom.

Queen Mary's Dolls' House is exhibited at Windsor castle. If you like miniatures and/or toys, you'd probably enjoy seeing it.

Windsor would get my vote, if only for the ease in getting to Heathrow in the morning.

Windsor is definitely closer and easier to get to. It's very inexpensive to take the bus from the terminal at Heathrow if you would rather not pay for taxi. It' drops you within easy walking distance to the castle. Windsor is a pleasant town to get around in. Also an easy walk over the Thames to see Eton if you have time.
I spend a lot of time at Hampton Court, also but maybe save that for a longer visit.

Windsor Castle no question.

I only have experience at Hampton Court, but I enjoyed it very much. Be sure to take a guided tour if you go to this site. Hopefully I can see Windsor someday.