There's riding available near Ringwod but not sure where. Tourist info should be able help (everywhere is full of tourist leaflets!)
We were in Mudeford last week which is about 7 miles from Bournemouth (to the east).
There's a Maize Maze near Sainsburys on the A35 which might be fun. We didn't get round to it because when the weather was dry the children only wanted to go in the pool/on the beach and when it was wet none of us fancied getting stuck ofr a couple of hoursi n a maze! More info here .
We also went to the aquarium at Bournemouth Time it carefully. Feeding of the sharks and rays is at 12noon and 2pm as well as, I think, 4.30pm. Our visit didn't coincide with these times at all so we missed out. And with two 6yo girls we were round in a little over an hour. Cost £18 for a family ticket.
Avoid the 'Hot Rocks' cafe which overlooks the aquarium and the beach as it is overpriced IMO and the service was lousy.
You have to get your children crabbing! This is the highlight of our holidays. 2 quid for a crab line, around another quid for a net and some bacon! We had four lousy days (well one hour each evening on those days) where we barely caught a thing, then on the last night, I stayed behind whilst my M&D took the girls and they caught around 30 in an hour!
Most of the things mentioned by everyone so far are within 10 miles of Bournemouth but if holiday traffic is bad, it may take a while to get around.
Have a fabulous trip.