One year later and I'm still looking for pieces.
Here is what I have. Philena (Stay) Bruette passed away in 1950. Her obituary says she was survived by her brother Ervin Bruette in New Hampshire. Philena is a Bruette by marriage.
Nancy, Philena's mother, married Joseph Bruette and Philena married Fred Bruette. Joseph is Fred's uncle. That would make "Ervin" Fred's cousin.
One of the more interesting notes on the 1930 US Census is the column for "age at first marriage." Nancy had married Arthur Atwood by this point but it says she was 16 at the time of her first marriage, presumably to Franklin Stay. I'm not even sure when Nancy was born but I believe either 1859 or 1860. That puts her marriage to Franklin around 1876, about two years before Philena was born. It fits the current information and supports the theory she married Philena's father.
Nancy married Joseph Bruette around 1886 when she was around 26. That means they divorced between 1876 and 1886. The 1880 Census isn't 100% clear but it says Nancy Stay was married at the time.
Any collaborating information would be appreciated.