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.
Previous answers discuss only one possibility that Donal may have been raised up to nobility by Robert Baratheon. One angle of Donal's name hasn't been discussed yet. That is, he could be a descendant of some minor impoverished knightly house. Westerosi are an old people and it has been 12,000 years since the First Men came to Westeros, followed afterwards by Andals and Rhoynar.
Many a great houses, noble and royal, rose in those 12,000 years. Similarly many a great houses fell in the same years. I would not discount this factor that Donal Noye might be descended from some landless, impoverished noble family. We do not know anything about Donal's family background nor do we know about any House Noye so no one can answer this for certainty. For all we know, House Noye may have been some part of ancient Stormlander nobility which, in current timeline, is diminished.
Examples of Low born people who shared same family name as some knightly/noble family are following:
Some Lords usually kick out their useless relatives out of their castles. It may have been that Donal Noye is descended from a junior branch of House Noye which was kicked out of their seat by the senior branch. Eventually, Senior Branch lost the seat (Because we don't know of any landed House Noye) and now members of the family are no different than low-born commonfolk. We have seen this trend in Merett Frey's account in Epilogue of ASOS:
So it is very likely that in two or three generations, Lords of Twins would start denying any kinship to their impoverished cousins from younger branches of House Frey. Eventually, these exiles would resign to live like smallfolk. Some may even become ironsmiths and swear service to some Lordly house. Some may find their way to Night's Watch. You never know.