Thank you so much for all the information that you have have posted in your messages regarding the Archdiocese of Boston's archives. I did send them an email yesterday requesting the two Ronan priests' mother's maiden name and the father's forename. In the meantime, I have found some more information online that will be very useful. According to the Massachusetts Vital Records (Death Index) 1841-1910, Mary Ronan is listed as follows:Ronan, Margaret(Donican) Boston 1888. Donican is a variable spelling of Dunigan so I am quite confident that she is my great, great, great aunt on my grandmother's side of my family.
I am very pleased to add this small family to my family tree. There is loads of information available online about the two Ronan priests on book and newspaper archives web sites. They are interesting people to research.