Whitby Museum, which has an excellent collection of local fossils. Visit if you haven't.
The cliffs to the north of Runswick are Wrack Hills; they're a site of special scientific interest which means you can't collect fossils from there; they were home a fine seam of Jurassic plant fossils. Above is one that fell from the cliffs in a block of sandstone. It may be only one left; in 1980, 1981, 1982 and 1984 Natural History Museum staff visited and excavated the cliffs, removing an enormous number of scientifically significant fossil plants without permission from the landowner.
Here's one that got away; looks like Ptillophylum pecten.
Reference: Palaeobotany and petrology of a Middle Jurassic ironstone bed at Wrack Hills, North Yorkshire. December 1985
Proceedings of the Yorkshire Geological Society,