I arrive at Heathrow at about 4 pm and would like to stay somewhere on the west side of the city close, or with excellent direct public transportation to Hampton Court Palace which I see the next day. I had thought about staying near Kew so I could visit and stay awake visiting the gardens after checking in. Or perhaps stay closer to Hampton. Any suggestions?
You can take the train from Waterloo Station to Hampton Court; it takes about half an hour. That would probably be the most convenient place to stay in London with a fast and easy connection to Hampton Court.
If you want to stay outside of the city, Richmond might be a nice option. You could take the bus from Richmond.
We recently started our trip in England with 5 days staying in Richmond. We went directly there from Heathrow by taxi. We took the bus to HCP on one of the days and visited Kew Gardens on another. It is a lovely area of greater London to visit and stay.
It would not make much sense to go into central London to come out again to somewhere convenient in west London.
Richmond would be a very sensible choice. You can get a 285 bus from Heathrow T2/3 to Feltham station for a 10 minute train ride to Richmond. Or take a cab from the rank to Feltham station and the train or a cab/car service all the way, about 12 miles.
There is a Premier Inn in Richmond a stone's throw (or shirt ride on the number 65 bus) from Kew Gardens, particularly th eLion Gate if on foot (the bus stops at all 3 gates).