Question about family and family ties. (Details inside)

Working with my cousin for several years on genealogy, including DNA testing.

She got a message through Ancestry from a woman in her late forties who also tested and as it turns out matches my cousin enough to be considered a second cousin (the child of a cousin).
The woman is very excited to have found someone she is closely related to because she was adopted and knows absolutely nothing about her birth parents or family.
Given the dates and upon reflection, my cousin has an idea who her birth mother might be. If she is correct, the mother is still alive, in her mid sixties, but no longer lives in the country where the child was born and given up for adoption.

This is something that occurred almost 50 years ago but I am sure the scars are still there. The child was probably conceived illegitimately and the young mother to be shipped overseas to hide it from family and friends. There has, as far as is known, never been an attempt to find the daughter and no mention of it within the family. If best to let sleeping dogs lie, how best to do so with the least pain for all? If she is interested, knowing my cousin, she (the lady who contacted my cousin) could trace my cousins family tree and make/draw her own conclusions so that kind of complicates things.

Where should my cousin go from here? Should she contact the person she thinks is the mother or lie to the lady that messaged her and say she has no idea? Is there a way she can give out broad medical and historical information without compromising the birth mother?

Any suggestions?