Danny Wilson is a younger brother of James Wilson. Danny started suffering from schizophrenia in his adolescent years. He later attended Princeton University and, even though he was on medication at the time, he still suffered from mild paranoia. As a result, he frequently spent hours on the phone complaining about his "mistreatment" with James who was in medical school at the time.
James tried to be supportive, but on one occasion he was trying to study for an exam when Danny called. Not having the time, James lost patience and hung up on his brother. The next day, Wilson's mother called him to tell him his brother had run away and left the meds behind. James blamed himself for Danny's disappearance.
When House started working at Princeton-Plainsboro, he found out they needed an oncologist and called Wilson. Wilson jumped at the chance. At first, he spent much of his free time visiting homeless shelters looking for Danny. However, apart from one sighting of Danny while James was sitting in a diner (when James couldn't leave fast enough to catch up to Danny), he has not seen his brother since he disappeared. He eventually gave up looking for him.
Wilson has generally been open to House about his relationships, but House had no idea about Danny until the episode Histories, despite the fact that House knew about Wilson's other brother and had met his parents. House was wondering why Wilson was so concerned about a homeless patient and Wilson finally told him.
Danny got into an altercation in New York and was taken to the psych ward of New York Mercy Hospital. At that point, hospital staff made the connection between Danny and James and let James know where Danny was. James made arrangements to see him and deliberately lied to and misled House to keep him from knowing why he was willing to miss monster trucks. Eventually, House realized something was up and pressed Wilson on the issue. To Wilson's surprise, House agreed to accompany Wilson to the hospital. At the critical moment, Wilson went in to see his brother while House once again was on the phone to his team with his latest revelation.
While Wilson offered to let House meet Danny, it never ended up taking place in the show.