To add to what people have said:
The Magpie does not have much of a queue at this time of year
Trenchers is now owned by a chain and the standards have slipped
On same road as the Magpie (50 yards before it) is the Quayside fish restaurant which I can recommend
There is not much happening at this time of year but the last weekend in October is Goth Weekend which is worth coming for. Unfortunately you would not find accommodation within 20 miles as it has all been booked up for months
For a laugh, a visit to the Dracula Experience is always good
The Lifeboat Museum (free entry) on the road to the pier is worth a look
The Whitby Museum in Pannet Park (open Tues - Saturday, nominal fee) is simply brilliant. It is so bad it is good. Where else could you see the severed hand of a criminal next to a display of knitted Spanish doll toilet roll holders? There is a whole section devoted to Captain Cook that contains nothing to do with him. For example, there is a spear from an Hawaiian island that Captain Cook may have sailed past