Well, if you are a history lover, then Portsmouth might appeal. It is the home of the Royal Navy and you can visit the Historic Dockyard, go on board HMS Victory, visit the HMS Warrior and view The Mary Rose which belonged to Henry VIII!
OK from Portsmouth, take a ferry over to the Isle of Wight and visit Queen Victoria's summer house, Osborne House, collect some coloured sand at Alum Bay, or hire a bike and cycle around the island!
Still not enough?
Hike along the South Downs Way. A 100 mile trail from Winchester to Eastbourne, or just pick a part of it to do, like the Hangers Way which is just 20 miles long.
Visit the New Forest and see the roaming ponies, wild deer and other wildlife. Visit Beaulieu Car Museum, run by Lord Montague or pitch up a tent by the river at Brockenhurst. Have a Cream Tea in Lyndhurst.
OK then, one more reason...
Visit the Cathedral at Winchester and roam around this ancient city. The Doomsday book was written here, Jane Austen died here and is buried in the cathedral. Visit Winchester castle, famous for its Knights of the round table.
If you want any more reasons, let me know!