What is a perfect family to Jessie?
Based on Jessie's account of her own life, she has never truly felt close with anyone besides her late father—not even her mother, whom she lives with and takes care of. When Jessie announces she's going to kill herself, she and Mama can't even get to the heart of things for a while—they're distanced from each other, easily able to say that they're more worried about things like towels and manicures than each other.
With that said, Jessie's perfect family would be a close one, made up of intimate relationships. Her parents would have gotten along much better, and her mother would truly care for her and never make her feel less-than because of her illness. Not only would she still be married, and her son be successful—they would know each other better than anyone else could.
Because so much of the play concerns Jesse’s inability to form an identity for herself, the issue is what sort of family would enable her to do this. She understands herself as having failed in personal relationships, and that her epilepsy has caused this, so she would think of “perfect” in terms of a family interacting with love and understanding as if she did not have this difficulty. Her son would not be a thief and involved with drugs, she would still be married to the man she loves and he would love her, her father would be alive, and her dog would be alive—and they would all give meaning to her life and therefore give her an identity she cannot seem to forge on her own.