Anne Hadley is Remy Hadley aka Thirteen's deceased mother. Her husband was John Hadley and in addition to Thirteen she had two other children we know about, Thirteen's now deceased older brother and Amy Hadley. She was seen in flashbacks in the episode Let Them Eat Cake. She was portrayed by actress Danielle Petty.
We first learn of Anne in the Season 4 episode You Don't Want to Know, when House does some digging after he sees Thirteen get upset for dropping a file on the floor. He finds a picture of Anne, but it's at least twenty years old. He figures that Anne is either dead or is estranged from Thirteen, and after a little more digging finds Anne's obituary and that she died after a lengthy illness. He guesses Parkinson's disease, but Thirteen admits it was Huntington's disease.
In Let Them Eat Cake, we learn that Thirteen hated her mother because of her mood swings, her poor treatment of Thirteen because of her inability to control her emotional state, and the fact she couldn't bring friends home because of her mother. She refused to accompany her mother to the hospital despite the urgings of her father that she would regret it. She admitted that she wished her mother would die. It was only after Thirteen reached adulthood that she regretted the way she treated her mother.