Southampton

In one word train.
Southampton is quite a railway hub and as well as London there are very frequent services to Bath, Winchester, Salisbury (for Stonehenge) and Portsmouth.
Also do get the ferry across to the Isle of Wight and get a day pass on the public transport there.

And there's a tourist information office on Civic Centre Road which could be helpful for more local places of interest. Portsmouth might be fun for your son, depending on his age. There is Admiral Nelson's ship, the HMS Victory, and the HMS Mary Rose, which sank in the harbour right under the nose of Henry Vlll, who had come to Portsmouth to see her launch.

Winchester is a short jaunt on the train and you can walk in about 10 minutes from the station to the High Street, and then to the cathedral. There is a very friendly tourist office on the High Street at city hall, down near King Alfred's statue (and very nice public toilets in the same building!). I would suggest you get a good guide book for this part of England so that you can really take advantage of this wonderful opportunity for you and your son.

Hi Jana,
I live not far from Southampton.
The city was heavily bombed during WW2 so although you can see some of the very old city walls (at the Bargates and a couple of other places) it's mostly 50's architecture that was rapidly built after the war.
There's a very modern shopping center, called West Quay, with some great shops.
The train station is an easy walk from the town center and also there's a free shuttle bus that will take you from the town center to the station.
As another poster suggested, you might like to take a day trip over to the Isle of Wight(pronounced 'white') on the Red Funnel ferry, which also within an easy walk of the town center, or again, you can use the free bus.
Depending on your son's interests, that might be a popular choice, it's quite an experience! I believe that ferry goes to Cowes; if you take a train to Lymington (you would change at Brockenhurst, but it's easy) there's another ferry which goes to Great Yarmouth on the Isle of Wight.
And another train journey could be to the seaside resort of Bournemouth, or you could stay on the train and go to Poole or Weymouth. Weymouth is a more old fashioned kind of seaside place than Bournemouth. Poole is a more of a port and has an interesting quayside.
And if your son likes cars, there's Lord Montague's car collection in Beaulieu (Bewlee) although I think you'd need a car to get there.
Well, it looks like I've written a book here! Anyway, if you decide to make the trip I think it's all pretty easy and there's certainly lots to do! best wishes, Maggie.