Dorset if you want the South - beaches galore, fantastic countryside, nice, historic towns like Dorchester, Sherborne, Bridport, Shaftsbury, easy trip to Salisbury for the amazing cathedral, and if you want really old, there's Maiden Castle outside Dorchester - the largest hill fort in Britain and bits of it are only 6,000 years old... you can even go to Portland to teach your children about genetics.
Northumberland if you want the North - 100 miles of beaches, endless countryside, hill and forest, more castles than you can shake a stick at, nice towns like Alnwick, Berwick, Rothbury and Hexham, Lindisfarne, day trips to Durham and Edinburgh, Newcastle for a good city day.
Anglesey if you want the West - huge numbers of small beaches, many with their own beach cafes, neolithic burial tombs you can go in, lots of good walking, easy access to the castles of North Wales and Snowdonia. Holyhead is a bit of a dump, but Conwy, Bangor and Caernarfon are nice...
slightly more off-field options - (1) Dumfries and Galloway, an area the size of the Lake District with 7 people and a dog called Colin in it. its got a 'book town', Wigtown, and some gorgeous seaside villages like Portpatrick. Dumfries isn't all that, Carlisle is nicer - if you need a city bit of your holiday, Glasgow is a looong way. (2) Inverness. nice, small city, surrounded by bucolic countryside, mountain, forest, the sea, beaches, nice towns like Nairn and Elgin, its only an hour out to Ullapool and the west coast. if you've not been it will take your breath away.