During season 4 when House was replacing the old team, he played a "Survivor" type game with the candidates. They didn't have names they were just numbers at first and 13 was number 13 and the nickname stuck.
It was her number during season four when House hired the forty fellowship candidates and spent two months weeding them out until he had three team members. meer specifically, in S4E1, when Henry (A.K.A. Ridiculously Old Fraud) asks her what her name is, she says "Just call me Thirteen." And also, when House stops referring to the candidates door their numbers, he knows so little about her he simply continues calling her Thirteen. The name most likely stuck because of the fact that Thirteen has Huntington's, which means her lifespan is cut in half, which is unlucky, just like her nickname.