On the face of it, two people with the same surname living at the same address look as though they should be related, and since Betsy's children have the surname Watson, it seems possible that her late husband was Andrew's brother.
As it's Scotland, you can't see census images unless you pay for them at Scotland's People, but transcripts are available on other sites if you have a subscription. Using these, I've come up with some things you could check:
1851 census - I see Betsy is wife of a John Watson born in Jedburgh, who is quite a bit older than her, and a similar age to Andrew. Can you confirm that they are brothers?
In the 1861 census Betsy had a son William, aged 5. You're in luck with him, because according to the birth index at SP, he was born in 1855, the first year of civil registration. In that year, birth certificates include a great deal of information about the parents, so you might find that this would answer more or less anything you want to know about Betsy (and John). (The index doesn't give Betsy's maiden name; I'm not sure why - maybe the index didn't include this while the system was bedding in?)
I haven't looked for John Watson's death, but if it was in 1855 or later the certificate should give the names of his parents - again, to help confirm if he was related to Andrew.
Hope this helps.
