Don't really think it's that weather dependent, although yes it's great if you have a sunny week.
The Whitby museum is great, even for children, so much to see and do. We were really impressed, right from children up to grandad!
There is a Whitby Wizard interactive science museum thingy. It seems really small from the outside but there is a surprising amount inside.
The steam trains are great, and in the summer they now go right from Whitby. Last year we went to Pickering, the kids loved the trains and the actual ride then we had a couple of hours in Pickering - coffee and cake, a walk around the shops and we visited Pickering castle.
The Abbey is lovely, and again good for a poor-weather contingency. There are interactive displays inside and a few activities like brass rubbing. The cafe there is lovely too - great cakes!
If you were really unlucky and it poured down every day, there is plenty to do a little way from Whitby, including other seaside resorts and interesting little towns.