Why does the author call the story "A Rose for Emily"?
It could be because of the description of the rose:
"When the Negro opened the blinds of one window, they could see that the leather was cracked; and when they sat down, a faint dust rose sluggishly about their thighs, spinning with slow motes in the single sun-ray."
It could have been that Emily was her daughter/friend who had also died, the same as the character in the book so he shows his respect by giving her the flower so that's why the book is named that way
There is no one answer to this question, and different theories have arisen throughout the years.
One story is that Faulkner wrote the story for a friend of his named Emily, and thus the story was a "Rose" for her (a gift, like the gift of a flower)
Another idea is that Homer is Emily's Rose. Often people preserve roses, allowing them to decay and dry up even after they have died, just as Emily has done with Homer.
Faulkner himself once said that the title came "from a picture of the strand of hair on the pillow", which is somewhat vague.
That quote from Faulkner was found on the enotes page linked below and is from the book: Inge, M. Thomas, ed. William Faulkner: A Rose for Emily. Columbus, OH: Merrill, 1970.