We have a 7 year old boy but are visiting our friends who have a 1 year old. From looking on Google we think we would like to visit Portchester Castle, Southsea Castle, Fort Nelson, Hayling Seaside Railway, Stansted Park, New Forest Wildlife Park, Royal Victoria Country Park, The Forest of Bere, the Local Haunts tour bus, Southsea Model Village, Little Woodham, Explosion! The Museum of Naval Firepower, D Day Museum. Are all these worth visiting?
Food wise are Azzurro, Las Tascas, Loch Fyne, Brasserie Blanc, The Old Customs House, Chimichanga, O Sole Mio, The Watermark and Dean's Diner any good?
Don't know the area too well, but when we have been: can recommend Portchester castle and Beaulieu in the New Forest with the Motor Museum which has indoor and outdoor bits and a monorail! Marwell zoo isn't too far either.
Porchester castle is good. There is a great play area in the woods at the west walk part of Bere Forest. Just pay to park. Loos and a snack van at the weekend. Fort Nelson is fab with all the tunnels and it's free. Gun wharf is worth a visit but avoid weekends. Lots of shops, places to eat and boats to watch. Marwell is great but pricey.
Oh please go to deans diner, it's great for a treat. Old customs house is good too but more pricey.
New forest wildlife park is ok but it's mainly forest animals. If you want a broader lot if animals go to marwell which might be marginally closer from Pompey. If you do go to wildlife park wear wellies and take lunch!
Castles are nice and explosion is supposed to be better than you would expect. they do a lot of events in there. have you comsudered the historic dockyard? they may have victorian christmas markets or similar soon.