The answer to your question seems to relate to the single strand of grey hair that is found on the pillow next to the corpse of Homer barron. The rather grisly discovery that the townspeople make in Miss Emily's house, that has remained undisturbed for so long, points towards a chillingly Gothic finale to this excellent story. Note how Faulkner concludes his tale:
Then we noticed that in the second pillow was the indentation of a head. One of us lifted something from it, and leaning forward, that faint and invisible dust dry and acrid in the nostrils, we saw a long strand of iron-grey hair.
This ending clearly implies that Miss Emily has spent the last years since the disappearance of Homer Barron sleeping with the corpse every night. Having lost Homer Barron once, she was clearly determined never to lose him again, and her murder of Homer Barron points to the way that through killing him she believed that she could retain possession of him. The strand of grey hair thus is an important motif that reflects these realities.
In William Faulkner’s story, “A Rose for Emily,” Miss Emily Grierson is painted as a very deprived woman. Growing up, she wasn’t allowed to have suitors; her father was extremely strict and protective. So, though she was beautiful, she was never courted, and she spent her teenaged and young adult life taking care of her father’s house. After her father died, Miss Emily met Homer Barron, a Yankee in town on business. Though Faulkner implies that most of the townspeople believed Homer was a homosexual, Miss Emily began to conduct a courtship with him. To the town’s surprise, the relationship seemed to be going smoothly. Miss Emily even buys a wedding present, which is a tradition among soon-to-be-couples. One day Homer leaves town, and everyone assumes it’s to prepare for marrying Miss Emily. Three days later, he’s seen returning to Emily’s house by the back door…and he’s never seen leaving.
The townspeople of Jefferson remember “all the young men her father had driven away, and … knew that with nothing left, [Emily] would have to cling to that which had robbed her, as people will.” And when you look at the story, she has lost quite a bit: she was deprived of her youth by her father, and then was deprived of her father, whose body she didn’t want to give up. The house her father left to her is in disrepair, but she continues to live there. And, all her suitors gave up on her, so as a middle-aged woman, Miss Emily wants to cling to Homer; he’s her last hope for the perfect life she believed she deserved, her last hope for something that would be all hers and never leave.
We as readers can’t know exactly what prompted Miss Emily to kill Homer, though again, one interpretation is that readers may be led to think Homer is gay, and we know from the story that Miss Emily herself is a demanding and snobby woman. Perhaps Homer tried to break off their engagement, or Miss Emily worried he would. The best clue we have is the poison Miss Emily buys—arsenic, which the druggist tells her is for killing rats. When we call people rats, it’s because they’ve double-crossed us, so most likely, Faulkner is implying that Miss Emily knew Homer could never truly be hers and that he would betray her love for him by not loving her back fully. Or again, perhaps he came back to Jefferson to try to end the engagement, which is why he tried to avoid being seen by coming in the back door rather than the front door. Either way, Miss Emily wants to be absolutely sure Homer Barron will never leave her, so she poisons him and enjoys many long years sleeping beside his placid corpse.