I found a Mary Bunker who was born at Eden, Maine, on June 3, 1841, daughter of Charles and Lydia (Lynam) Bunker. Lydia and her children were living at Rockland, Maine, at the time of the 1860 Census; Charles had died in 1858.
On September 30, 1865 at Rockland, Maine, according to a recreated Maine Marriage Record, Mary Bunker of Rockland, Maine, married A. B. Jones, who was then living at Boston, Massachusetts. Interestingly, I found a Mary B. Jones, a milliner, living in the household of hotel keeper Alexander B. Jones at Castine, Maine, at the time of the 1880 Census. And, fittingly, there's a Maine record for the October 26, 1898 death of Mary (Bunker) Jones, the same daughter of Charles and Lydia - at Castine, Maine.
However, in the Census description, Mary was described as Alexander's step-daughter, and a Lizzie E. was described as his step-granddaughter. Alexander was presumably Alexander Bubier Jones, son of Daniel Illsley Jones and Elizabeth (Bubier) Jones.
It's possible that Lizzie E. Jones was the future wife of Walter A. Low, son of Sheriff Augustus T. and Eliza Low of Rockland, Maine. And it's possible that Lizzie is the person who wrote the identification.
If you have corrections and/or additional information on the Mary (Bunker) Jones described above, or an alternative theory as to the identity of Mary B. Jones, please leave a comment or contact me directly.