Mike and I stayed this time at the Arches guest house on Hudson Street.

http://www.iknow-yorkshire.co.uk/accommodation/1124-arches_guest_house-whitby.htm

£28 per person, per night sharing a double room in our case. They're very friendly, comfortable and clean, and do a great breakfast.

I've also stayed in two cottages and at the Avalon on the Royal Crescent and have no complaints about any of them. There are loads of other alternatives though.