She has no back story, family, or a last name, but her solo episode makes the issue of old age care affect the audience.
Sophia finds out from her friend Gladys in Shady Pines that Lillian was transferred from there to Sunny Pastures which she is starts to claim the worst of the worst of Nursing Homes, because of her mother's reaction, Dorothy decides to visit the home as well and Dorothy herself was shocked but how bad it was. Sophia wants to visit Lillian often as she can but she decides to break Lillian out of the home which she does with Rose.
The next day Dorothy tells the staff of the home, that Lillian is visiting, it's clear Sophia is tired as Lillian is hard work and it's realised that Lillian is Senile and needs special care. Dorothy and Sophia meet with the manager of Sunny Pastures however he is unable to help due to lack of funding and the problems of running the place. When they return Rose reveals she has found a home that will help Lillian however she will need an extra $150 a month as her own benefits don't cover it, but it's Blanche that saves the day, she gives her bonus that she was going to use for Breast Enlargement and help Lillian.
Lillian's story has a bitter-sweet ending as Dorothy remarks that while they helped one woman Sunny Pastures is unchanged, but as Blanche reminds Dorothy that Lillian was lucky, there is probable more people worse off, this was the episode's meaning, retirement home horror stories have become more common and state run ones have been in the news due to negative reasons.