If you're doing a gentle explore around Northumberland it's well worth popping to Wooler for a couple of hours. Seahouses is well worth visiting with the kids (take plenty of 2p and 10p pieces for the amusements) and is a stones throw from Bamburgh Castle.
If you're heading out towards the west of the county Hexham is a lovely little town to explore. Nearer the south-east of the county there's the fantastic Woodhorn Colliery Museum which not only has a lot about the mining heritage of the area and the fantastic Pitmen Painters, but they also tend to have different non-coal mining displays and events that change every week or two.
If you're in that area then stop off at the Gingerbread Coffee House in Red Row for some lovely home made cakes to go with your tea or coffee. That's close to Druridge Bay which is another great place for kids provided its not raining, also have the QEII Country Park just along from Woodhorn.
If you're into your history there are a ton of old castles and peel towers many of which are accessible to wander around and explore. Warkworth castle is an excellent way to kill an hour or two if you're into ruins, though as already mentions Alnwick Castle and Gardens can fill the majority of a day if you're around there.