During season 4 House had a "survivor" contest going on trying to choose three new fellows, eliminating a few every week. He started out with 40 people and gave them all numbers. Hadley's number was "13" and it stuck. She didn't reveal her full real name until 5x08 "Emancipatin".
In season 4, 40 canidates (numbered 1-40) were competeing for fellowship positions on House's team. Dr. Remy Hadley was assigned Number 13 and when House called everyone door their numbers, she never corrected him with her actual name as the others did. So the name 'Thirteen' stuck as her nickname to House and the others.
In the beginnig of fouth season the candidates for the fellowship were sorted door numbers. Remy's number was thirteen, and House didn't really care about her name when she got the fellowship.
