The rose in the title can be interpreted as a tribute of respect for Miss Emily who has lived a life dominated by her controlling father, the social requirements of the South and circumstances beyond her control.
Miss Emily grows up in the waning years of the great South, the plantation society that her father dominated prevented her from marrying. He found all her suitors unacceptable, beneath her socially, rejecting one then another. Miss Emily is thus left alone, with no husband, no family.
Her father's death frees her to a degree, although a little late for her to capture the joy that she was denied. However, she begins to go around with Homer Barron, the Yankee laborer. This behavior creates quite a scandal in the town, whose reaction results in the calling of Miss Emily's family in another town to come and stop her from making a terrible mistake. The town is perched on the edge of their collective seat waiting to see Miss Emily marry the Yankee, an act of true degredation.
However, they are surprised to learn, or so they believe, that Homer Barron has left town, never to be seen again. Miss Emily grows old in her house, alone, except for the servant who assists her. When she dies, the town can't wait to get inside to look at the remnants of the recluse, Emily's life. They are however shocked to discover that Homer Barron never left town, but was murdered and his body placed on a bed where it rotted and decomposed.
Miss Emily, it appears, slept next to the remains of Homer Barron, as evidenced by the gray hair found on a pillow.
I think that the rose in the title is a way of expressing both respect and remorse for Miss Emily's life, a life dominated by death, loss and loneliness.