Down the road from Longdowne Farm is The Otter and Owl Centre. It is my dc's favourite place to spend a few hours when we visit my parents (although I see it has changed it's name)
Mudeford is lovely for crabbing and you can get the ferry over to the beach or drive to Hengistbury head and get the ''noddy train'' (as we call it) to the beach.
Bucklers Hard is quite pretty and close to Beaulieu. Exbury gardens good if you like that sort of thing.
If you do go to Lymington, avoid a Saturday, unless you like markets and queueing for car parks.
You can get the ferry from Lymington to the Isle of Wight for the day.
The Mayflower Inn is a nice pub in Lymington. As is The Hobler on the road in to Lymington.
Hurst Castle is a trek if you like walking. Keyhaven (and the Gun pub) is a pretty place nearby to see it from without walking all the way down the spit!
Marwell Zoo is doable for the day too.
And, of course, there is all the beautiful forest for picnics, playing hide and seek and tree climbing!