Though relatively minor characters in the story, the eponymous sisters have great significance in relation to the story's outcome. For it is through the sisters that we find out about the details of Father Flynn's shocking mental and physical decline in the time leading up to his death.
In imparting this crucial information the sisters elaborate on the vague statement made by Old Cotter that there was something peculiar about Father Flynn. Apparently, the duties and responsibilities of the priesthood were getting too much for him. The psychological stress of carrying out his priestly duties was so great that, one day, he broke the chalice. From then on, it was all downhill, and Father Flynn was never truly the same after that.
Father Flynn's sister Eliza has the last word, which is instructive. In telling the narrator and his mother the terrible truth about Father Flynn's decline she is acting as a kind of gatekeeper to his memory, putting forward a version of events which she hopes will be canonical. Thanks to Eliza and her sister, from now on, when people think of Father Flynn, they will remember Eliza's vivid description of the old priest laughing softly to himself in the dark confessional.
While the narrator, a boy, is clearly the protagonist of James Joyce’s story, it is the dead priest’s two sisters who play the largest roles in helping him understand some essential elements of the priest’s life and what mourning and grief may entail. The gap between the boy’s understanding of death and its rituals compared to the devotion of the sisters, Nannie and Eliza, is first suggested by their prayers over their brother’s body, which the narrator hears as “mutterings.” The sisters also play distinct but complementary roles: Nannie takes care of the hospitality aspects, serving the guests sherry, while Eliza recounts stories about her brother. As the boy and his aunt sit and listen to Eliza’s comments, he realizes that she needs to unburden herself of information about her brother’s last months. Rather than reflect on his long term of positive contributions, Eliza reveals the difficulties he faces as his mental health had deteriorated, which impeded his ability to function as a priest. The interior space of their home and the close personal relationships the priest had sustained are the place of mourning, which is deeply felt on a personal level, in contrast to the apathy the boy had noted in the sunny exterior world.
In the opening paragraph of this story, the first in the collection Dubliners, James Joyce includes three words that serve as themes for the rest of the book: paralysis, simony, and gnomon. While not all three ideas are relevant in each story, they are all relevant here.
However, to answer this question, I’d like to focus on the idea of a gnomon. Simply put, a gnomon is what’s left of a parallelogram when a similarly-shaped parallelogram is removed from one of the corners.
But what does this have to do with the title “The Sisters”?
Symbolically, a gnomon represents the idea that something has been removed and is being left out; the story is complete, but incomplete at the same time. In “The Sisters,” Father Flynn is an adult the unnamed narrator looks up to and spends a lot of time with. However, he doesn’t truly understand the depths of the old priest’s mental afflictions until he hears the father’s sisters talk.
As stated in a previous answer, once the boy realizes the extent of Father Flynn’s mental deterioration, the old priest’s hold on him is removed. For the boy, the partial shape—the gnomon—becomes the whole.
This short story, which is the first in Joyce's collection of short stories entitled Dubliners, captures the complexity of life and explores the character of the dead priest, Father Flynn, who is examined through the testimony of three characters. This story, as with all of the short stories in this collection, is based around an epiphany, when the central character experiences a sudden revelation or moment of self-understanding which allows them to grow up or mature, passing some kind of ritual allowing them to move towards adulthood.
One reason that the title "The Sisters" was chosen is that it is the testimony of one of Father Flynn's sisters that allows the narrator to go through this rite of passage and achieve his epiphany. The reader is initially presented with the narrator's own remembrances of Father Flynn, which are nostalgic, then the suspicions of Old Cotter about Father Flynn, which put him in an altogether sinister light. Finally, it is Eliza's account of his final days that allows the reader to see the mental pressure he was under and how that became too much for him. Note what Eliza says about her deceased brother:
He was too scrupulous always, she said. The duties of the priesthood was too much for him. And then his life was, you might say, crossed.
It is at this point that the narrator is able to approach the table and taste the sherry, a clear copy of a first communion ceremony, that indicates how important this moment is as a rite of passage and signals the epiphany. The short story is therefore called "The Sisters" because it signals the importance of Eliza and Nannie, the sisters of Father Flynn, in revealing crucial information about Father Flynn and his character, which leads to the epiphany of the narrator.