St Nicholas's Hospital was a London hospital important to the Tenth Doctor because of events there on late Christmas Eve 2006. On that evening, a young boy named Daniel Francis Thompson struggled for his life in the A&E department, after a car accident at around 9:30pm.
Somehow, as the boy lay in a coma, the Doctor's TARDIS had been pulled into the boy's subconscious struggle for survival. In a way real only to the Doctor himself, the two made a tour of various Christmases of the past. Once the Doctor deduced that Dan wasn't in fact physically present in the TARDIS, he sourced Dan's true location to St Nicholas's, and instructed the mental projection of Dan to try to wake up.
With the projection of Dan now out of the TARDIS, the Doctor was free to make sure Dan's chances of survival were optimised. He thus tipped the balance of fate a little bit more in Dan's favour by ensuring the ambulances would hit all green lights, that the hospital had adequate blood supplies, that usual cases of Christmas drunks were kept away, and the like. He even made himself a familiar figure in the hospital, popping back in time six months so as to establish himself with members of staff.