At the tail end of an article explaining Irish naming traditions, one website providing research tips explained the key about who was named as godparents. For "the poorer class of Catholics" in Ireland, the naming of godparents followed one simple rule: each godparent was "either a sibling or a sibling in-law of one of the child's parents."
If that is so—and may provide clues as to who the father of this mystery Dennis Tully I've found might be—then let's take a look at the possible roster of Tully siblings by examining whom our own Denis Tully named as godparent for his children.
Denis Tully and Margaret Flannery—our Denis, as opposed to the younger Dennis found in the family tree of some DNA matches—had at least seven children born to them in Ireland. Of course, there may be more. Some may have been stillborn, something I wonder as I see gaps in the dates of birth for the others. Some may have been born to Denis and Margaret prior to the oldest child I've found in baptismal records, as previous baptismal records don't seem to be available before 1832. Those are possibilities to keep in mind. But for now, let's look at the ones I've been able to find.
The baptismal entry for eldest child Johanna, baptized in March of 1832, was a challenge to read. Her first name was entirely broken from the page. The only reason I know it was hers was because a descendant of Johanna shared a baptismal verification letter with me which matched, almost word for word, one which the Tully family had passed down through my father-in-law's line for younger brother John. With the crumbling state of that old record, it is not surprising to discover that the entry for the sponsors was also hard to read. The Tully godfather's name showed only the last two letters: k and e. Luke?
The next child, Michael, was baptized on June 5, 1834, with two Flannery relatives serving as godparents: William and Bridget.
Third-born Patrick was baptized on August 30, 1836. His sponsors were Thomas Tully and Mary. Once again a hard-to-read entry makes it difficult to determine Mary's maiden name, but my guess is: Tully.
The next child, ill-fated William, was baptized on May 8, 1839. Once again, a Tully name shows in the godparents' listing: John Tully, along with Judy McNamara.
Following him was son John, baptized on February 24, 1842. While his godfather's name was difficult to read—John Brun?—neither sponsor was a Tully, as the godmother was Mary McNamara.
Daughter Margaret, next in the birth order, was baptized in September of 1844. Her sponsors' names provide me with what I consider the best possibilities: Mick Tully and Mary Gleeson.
The last child born to Denis and Margaret in Ireland they named Honora. She was baptized on February 25, 1847. Like William before her, she apparently did not live long enough to make the trip across the Atlantic with the rest of the family. Once again, John Tully shows up as godfather for this child, along with Biddy Tully as godmother.
After that, the family made the voyage to Canada, and their youngest child—whom they named William, providing the clue that the earlier son by that name had not survived—was born about 1850. Although I have been unable to find his baptismal record, it is possible that immigrants in Canada did not always follow the customs from their homeland in identifying sponsors for their newborns.
With that task, I now have a list of Tully brothers who could have been father to the younger Dennis. Based on what I've found, among the names to seek in baptismal records would be Tully fathers named Luke, Thomas, John, and Mick (or Michael). Considering traditional Irish naming patterns, seeing Denis and Margaret naming their own firstborn son Michael convinces me to look first at records for a family born to someone named Michael or Mick Tully.