In Game of Thrones/A Song of Ice and Fire, bastards have the last names such as Snow, Sand, Flowers, Hill, Pyke, Rivers, Stone, Storm, Waters. In the event that they are never legitimized what happens when these bastard children become adults? Do they and their offspring forevermore hold these last names or as they grow up do they invent a name for themselves? Would that even be allowed?
Edit Pt2: What happens to unacknowledged bastards? Do they get to make up names too?
Bastard names are given only to bastards with at least one parent of high birth. So the bastard child of two peasants would have no surname at all.
Thus a bastard name like "Snow" or "Rivers" is simultaneously a stigma and a mark of distinction. The whole thing with bastard names is custom, not law.
The highborn parent can bestow the usual name, a new one of his/her own devising, or none at all. Most legitimate sons of bastards keep the bastard name, but there are cases where a later generation fiddles with it to remove the taint. There's one such case that you will meet in the next book, a minor character descended from a Waters (a bastard name along the shores of Blackwater Bay) whose great grandfather changed the name to Longwaters for just that reason.
Only a highborn has a surname. A bastard of a highborn receives a bastard surname such as Snow, Flowers, Sand, etc.
So we have Jon Snow. If he marries a peasant (provided that he doesn't leave the Night's Watch) it doesn't matter what his child is named as he is a highborn bastard and a peasant doesn't have a surname, so who cares.
On the other hand, if some dashing highborn maid fancies him and is soooooo savvy to convince her parents to let her marry Jon (either so her family can gain prestige from being married to a Stark descendant or just because... ) in this case Jon could take the name of the girl's family. He would become noble and holder of lands. Therefore he would be permitted to adopt the name of the family or invent something.
That happened to that bodyguard of Tyrion (Bron)? He married into nobility! Pay attention, he is now known as Lord Bron of Blackwater (he invented a name for himself after being betrothed to Lolys).
As far as I can tell from the piece on bastardy at AWOIAF, you only get to change your name if you're legitimised by royal decree; otherwise, you're stuck being a Snow/Pyke/Flowers/etc for the rest of your life and everyone will always know you're a bastard e.g. Jon Snow, despite being acknowledged by Eddard Stark
retained the bastard name of the North and the social status it conferred
What happens to their offspring is less clear, and I can't think of an example in the books where it's occurred. As smallfolk don't have last names at all, I would say that the child of a bastard would follow this pattern and not have a last name.
First off, only the highborn bastards received their special surnames that is to say to those who are at least acknowledged by their highborn parent(s) or claim to be bastards of some noble e.g. Donnel Hill or Glendon Flowers. The unacknowledged bastards are no different from commonfolk who have no last names e.g. Gendry.
If a bastard is not legitimized, they will carry their bastard surname for their entire life. If they are legitimized, they adopt the name of their parents e.g. Ramsay Bolton or form their new cadet Dynasty e.g. Daemon Blackfyre.
If a bastard sires trueborn offspring without getting legitimized themselves, their Children are not considered bastards and do not have to carry the bastard surnames. They can choose their own name. E.g. Reniffer Longwater's ancestor.
"I see you wonder, what sort of name is that?" the man had cackled when Jaime went to question him. "It is an old name, 'tis true. I am not one to boast, but there is royal blood in my veins. I am descended from a princess. My father told me the tale when I was a tad of a lad." Longwaters had not been a tad of a lad for many a year, to judge from his spotted head and the white hairs growing from his chin. "She was the fairest treasure of the Maidenvault. Lord Oakenfist the great admiral lost his heart to her, though he was married to another. She gave their son the bastard name of 'Waters' in honor of his father, and he grew to be a great knight, as did his own son, who put the 'Long' before the 'Waters' so men might know that he was not basely born himself. So I have a little dragon in me."
AFFC - Jaime I
