Thirteen is one of many doctors temporarily hired by House with the intention of gradually screening them out. She is called Thirteen because all the candidates were originally given numbers. However, in the episode "Mirror Mirror" it was suggested that her full name was Remy Hadley. During "House's Head" her identity is confirmed by Lisa Cuddy as Dr. Hadley. Finally, in "Emancipation" she identifies herself to a stranger as "Dr. Remy Hadley".
In the episode "The Down Low" (season 6 episode 10), it is revealed that Thirteen's middle name is Beauregard. Therefore, her full name is Remy Beauregard Hadley.
In the episode "97 Seconds," she got distracted and failed to give a patient his pills, resulting in both his death and the death of his service dog, who ate the pills off the floor. House fired the rest of her team, but kept her, arguing that she would never make such a mistake again.
She was at odds with Amber, the only other woman remaining on House's crew. Thirteen was hired along with Kutner and Taub.
In the episode "Don't Ever Change," there are a few little hints that Thirteen may be bisexual. In the episode "Living the Dream," there is confirmation for these assumptions when Thirteen, Foreman, and House are watching a soap opera, and she has a line, "I think I dated that Nurse... (pause)... no."
One of the things which originally made Thirteen "interesting" to House was the sheer lack of information she gives out about herself, so he began trying to find out as much as possible. In "You Don't Want to Know," he figured out that she is the only one on the team who drinks decaffeinated coffee, and he replaced all her coffee with regular. Thirteen overreacted to her resulting clumsiness, and House found out from her that her mother died of Huntington's chorea, a hereditary disease with the early symptom of physical clumsiness.
Thirteen did not deal well with the knowledge that she might have Huntington's. She refused to get tested, and when House tested her without her knowledge in "You Don't Want to Know," she refused to look at the results. She believed that by not knowing, she was giving herself the courage to do things she would never have been able to do otherwise. However in "Wilson's Heart," the death of Amber, another young female physician, led her to self-test. The result was positive: she has Huntington's.
In "Lucky Thirteen," Thirteen is on a destructive downward spiral. Confirming her bisexuality, she's found partying with a girl whose name she doesn't know. The girl begins to have seizures, and so Thirteen has the girl brought to Princeton-Plainsboro. During the episode, Dr.Cuddy demands that Thirteen take a drug test after finding her giving herself fluids via an IV to deal with her hangover. House saves her from the test, but fires her instead. By the end of the show, Thirteen is rehired -- House was only toying with her emotions. The episode closes with Thirteen continuing with her destructive habits and partying with another woman.
In the 5th season, Foreman gets her into a Huntington's clinical trial and in the finale of "Joy to the World", the two of them kiss. They eventually start dating. In the clinical trial, she was getting the placebo instead of the real medicine. Foreman found out and gave her the actual medicine. As a result, she developed a brain tumor which left her temporarily blind. After radiation treatment, she was fine.
In "Help Me", Thirteen requests a leave of absence for personal reasons. However, there is one scene in the episode where she is holding a drink and her hand shakes -- Thirteen's disease, Huntington's, slowly affects motor control as it progresses.