There is no mention of a House Noye on AWOIAF, and besides the fact that Donal Noye served at Storm's End, no indication that he is of noble birth. Does anyone know why he has a surname?
Reasons Donald Noye had a surname in Game of Thrones(theories)
Noye helped the king. Maybe the king offered him knighthood, but he turned it down and instead asked for a name. He did show bravery and other knightly qualities, so I think it can be said he was offered the knighthood. But something stopped him from achieving it.
Noye did "great" things and he was rewarded for those things by being given a name.
This quote was supplied by westeros.org, but is apparently from Chapter 19, A Game of Thrones:
This is sort of connected to my above theory. And it was said by Stannis so that sort of talks about a godly interaction or that it just wasn't meant for him. I mean being a lord commander or being a knight.
According to the wikia:
So what would be the use of knighting a armless "knight", its not like he could do anything. So instead he was given a name. This theory, however, has some flaws.
Who can knight you so that the knighting would stick? A SFF.SE question.
Alternately(from the westeros.org link) the user RumHam supplies:
This is pure guesswork, but on A Storm of Swords, Chapter 76 (Jon) we learn that he used to work as a blacksmith at Storm's End. There he forged Robert Baratheon's warhammer, which was used to crash Rhaegar at the Rebellion, which led to victory.
So, maybe Robert rewarded him, raising him in the society, by giving him a last name or allowing him to choose one.
I haven't seen any official source explaining why Donal has a last name, so unless Martin tells us, it's just guessing.