In the episode S08E03 of Game of Thrones, the plan of the defenders for vanquishing the Night King is to lure him near Winterfell using Bran for being able to approach him and kill him. It is well explained why the Night King wants to eliminate Bran: he wants to erase humans with all of their knowledge. Bran is the Three-Eyed Raven and as such the memory of the world. By being alive he is a threat to the Night King's plan.
But why would the Night King want to kill Bran personally whereas any minion could have done it? Especially when Theon was on the verge of being submerged by the wights and there was apparently no need for the Night King to kill Bran in place of any wight. Moreover because Bran is marked, the Night King knows at any moment where Bran is, so he would certainly know when Bran dies, making his presence useless for checking that Bran is indeed dead.