As Dick has said, you effectively only have one day, so I would try to eliminate too much travelling around.
Brighton is great for history, music and pubs and it’s a 2 hour direct train ride to Southampton. It’s surrounded by the South Downs National Park if you want countryside. October is a great time to visit. Brighton is easier to get to from Gatwick rather than Heathrow. Where are you arriving into?
Windsor is an easy hop from Heathrow and would meet your criteria. From Windsor, take the train to Southampton in c 1 hour 45 minutes, changing at Reading.
Winchester also meets your requirements and is a short 20 minute train journey to Southampton, however, getting there from Heathrow would either involve taking a train into London, the underground then another train or taking a bus from Heathrow to Woking, then a train. You may not feel like either option with jet lag.
Alternatively, you could stay in Southampton. Whilst Southampton itself isn’t that exciting (it was heavily bombed during WW2 and has been rebuilt), it would allow you to take a day trip into the nearby New Forest and visit Beaulieu House, look out for the wild ponies that graze the heathland or visit the Isle of Wight. Portsmouth isn’t far away if you want naval history.