2 weeks! yes, long time for little ones Are you cottaging (!) or BB or camoing for 2 weeks?
Anyway, some thoughts for you.
1: Keilder forest - huge great area of beuty with woods, huge lake with boating etc, picnic areas.
2: Alnwick castle has harry Potter kudos (exterior shots but down is now filled with HP fans and 'souvenirs' It's a pleasant bustling market town anyway.
3: Banburgh castle - with beautiful, huge clean beach
4: Lindisfarfne (Holy Island) well worth a trip for te fun of drving accross the 'sea' - it's a tidal island, so make sure you check out times to get across, and more importantly back again.
5: Stephenson Railway Museum (abou 25 miles away,but you can have a ride on a steam train.
6: Beamish Museum (45miles away in Stanley) is a fab day out, it's a living history outdoor museum about life in the north east with trams to ride on, olde fashioned shops, streets, houses, school, mine. Quite expensive but a lovely day out for all ages.
7: Go to newcastle for a day (30 miles) Hancock museum worth a visit while you're there, very central. Loads of cafes, shops etc 5 mins walk away from museum.
8: If you want a junky day, go to Metro centre in Gateshead (again, 30 miles) with shops, bowling, multiscreen cinema, fun fair, laser quest - you get the idea!
9: Cragside is near Alnwick, beautiful house with acres of stunning gardens
10: Not sure how far Morpeth is from Alnwick but not too far (Probably about 30-40 miles) but this looks good
Hope the weather is kind to you