Thank you all so much. Every little bit helps and Gadget that find of yours - Matthew Cumming being at Woodside is great as Woodside is as near to Burn Hills Farm as is Stephen's Hall.
Now why didn't I think of there being other Stawpart's at Stephen's Hall? Though the confusion over the father of Mabel being William or Thomas is a bit odd. I later found that there was another birth before 1766 - Cuthbert Stawpart in 1761.
The reason I thought it had to be Elizabeth Oliver who was the mother of Mabel Stawpart was this bit by Geoff Nicholson in his history of the Stephenson's of Stephen's Hall (at a later date) : "William Sr had married Mary nee Stokoe (c1743-1793), widow of Joseph Stawpart (-1766), the previous tenant of Stephen's Hall, who had a daughter by his first wife and one by Mary."
Since Joseph married Elizabeth in 1746 and she died in 1762 I jumped to what looked obvious without thinking. It could be that Joseph's first wife was before Elizabeth, or that daughter's baptism is missing or I missed it.
Another interesting point is that in another account by Geoff Nicholson on the Stephenson Family who once posted this:
"Alternatively, a good possibility for Robert's parents would be Anthony Stephenson and Elizabeth Oliver of Burn Hills Farm, near Ryton. They would seem to have arrived in the Ryton area about 1789 where the baptism of their second child and ensuing children are noted in the Ryton registers. "
This is of course a different Elizabeth Oliver who married Anthony Stephenson in Ryton on 14 May 1786. I think she was the daughter of Nicholas Oliver who was the son of William and Elizabeth Cumming. So perhaps that is how the Stephenson's acquired Burn Hills Farm. That would have given them a large expanse of land between Stephen's Hall and Burn Hills.
There is a fantastic description of Stephen's Hall in the Northumberland and Durham History pages. Briefly it dates back to early 17th century, has a number of Latin Inscriptions, and it dates back to 1635 when a Stephen Coulsin or Coulson lived there. A rare wall painting was found and even an old spice cupboard. We are told that it is worth visiting, but when I tried Google satellite pictures and then street view it is cluttered with motor homes and offers Xmas Trees for sale.
Thank you all again, Malcolm