Hi,
Based on the information you have given, I have found the following:
The Eleanor Hay you mention in your post is described as born c.1836 in Norham in the 1871 census with her sister Agnes.
In the 1841 census, there is an Eleanor Hay of around this age living in Norham with her father, John, and her mother, Eleanor:
https://www.familysearch.org/ark:/61903/1:1:MQBH-MC6The youngest child living with them in this census is George, who is 9 months old. There is this corresponding birth registration:
HAY, GEORGE FISH (mother's maiden name)
GRO Reference: 1840 S Quarter in THE BERWICK UPON TWEED UNION Volume 25 Page 22
So this seems to be the right family.
Queenie
