Hi Nicky
I live in Redcar so can contribute a little to this, I'm sure Jane will be able to add some more when she's online, I think she's well up on cemeteries in the area.
There is a cemetery at Christchurch, Coatham Road, which may well have your ancestor's grave, although it's quite small. Redcar cemetery has apparently been open since about 1874 so that's another possibility, that one is a very large cemetery.
Jane advised me to contact the cemetries officer for Redcar when I was looking for a burial, I did and he was very helpful, here is the link
http://www.rootschat.com/links/0wx/