If we don't want to do big-city London touring on this one-day layover (arr Heathrow 8:30a, dep Gatwick 7:30a next day), what suggestions do you have for a nice day and overnight in the surrounding countryside using public transport?
One of the coastal towns easily reached by train from Gatwick. Brighton or Eastbourne, come to mind and then stay at one of the airport hotels as you leave so early. Leave luggage at Gatwick. I have used Premier Inn, Courtyard hotel and Sofitel for pre-flight nights.
You don't have much option other than to stay at Gatwick, as you will need to be at the airport at 4.30/6.00 am depending on where you are heading.
Brighton is the most obvious for a day trip, but it's urban, not countryside. There is countryside surrounding Brighton, so you could head for the Downs.
It will take you until lunchtime to get from Heathrow to Gatwick, so you won't have as long as you think available for sightseeing.
Not sure when you say "don't want to do big-city London touring," do you mean you don't want to see London, period? Or you don't want to try to see all of London at a glance? If the former, this list of Things to Do Near Gatwick Airport may give you some ideas.
https://www.tripadvisor.com/AttractionsNear-g186338-qLGW-London_England.html
Thanks for all the responses. We are renting a car and going down to Beachy Head for the day. Found a lovely B&B in Gatwick's backyard so we're set for the morning. Check out Latchett's Cottage in Horley!