Winchester is fairly close (Jane Austen is buried there in the cathedral, and they have King Arthur's round table (actually a 13th C table with 16th decoration)
Salisbury is about an hour by car or an hour and a quarter by train - they currently have a 'Barons Trail' as part of the Magna Carta 800 celebrations - there are lots of model Barons around the town, decorated in different themes (there is a Terry Pratchett Baron, for instance) and of course there is Magna Carta itself, too.
I agree with the suggestion to visit Beaulieu Motor Museum - as well as the cars themselves (which are lovely, even if you are not a petrolhead) there is the ruined abbey and the manor house to visit.
I'd pick Avebury over Stonehenge (particularly with kids - there is much more space and you can explore more), but athersgeo is right that it would take a while - I'd say around 1.45 - 2 hours drive each way.
If you/your kids are interested in history there is Fishbourne Roman Villa, about 20 minutes from Portsmouth, and there are Palmerston forts aabove the city https://en.wikipedia.org/wiki/Fort_Nelson,_Hampshire