Cragside is great and has a good playground to have a picnic in (you just have to find the right car park). Expensice though if you are not a National Trust Member.
Alnwick garden OK, but pretty expensive too.
Some of the English Heritage places are good value for money. Warkworth Castle kept my children entertained for quite some time and Warkworth is lovely for tea and cakes and a walk along the river.
Dunstanburgh Castle good too. However, it is quite a walk and depends how your kids are with that. If you park in Craster its about 1 1/2 miles (very pretty though along the coast).
There is a Barley mill at Ford which is nice and a little train which goes about a mile to an English Heritage castle at Etal (and you can have tea outside at the post office).
Conundrum farm just outside Berwick. Big indoor barn with little tractors to ride around and then you can buy feed and walk around and feed the animals.
If its raining, Potadoodledo just outside Berwick is good for painting pots and a cup of tea.
Well worth the drive over to Holy Island. After you turn off the A1, there is a cafe about a mile down on your left hand side. Much nicer (and cheaper) than anything on the Island. It has beautiful views and a little playground.
If its sunny, you can't beat the beach. My favourite spot is at Bamburgh. Go into the village and (assuming you are driving from the A1 and not Seahouses), turn left before you get to the Castle. There are some little car parks down there and a fab bit of beach with some tidal pools that are safe for little ones.
As you can see - I have spent a lot of time in Northumberland with small children.