Lord Brynden Rivers AKA "Bloodraven" was a legitimised bastard son of Aegon IV Targaryen.
My question is, Why did he keep "Rivers" as his last name and not change it to "Targaryen"? or at least "Blackwood"?
Consider this line from A Storm of Swords:
All he had to do was say the word, and he would be Jon Stark, and nevermore a Snow.
-A Song of Ice and Fire: A Storm of Swords, Chapter Seventy-Six (Jon).
As others have pointed out, the only examples from A Song of Ice and Fire of legitimized names are (hypothetically) Jon Snow becoming Jon Stark and Ramsay Snow becoming Ramsay Bolton. The Princess and the Queen adds another example: Addam and Alyn Velaryon, the bastards sons of Lord Corlys Velaryon.
Since none of the Great Bastards went by Targeryen, it clearly isn't a concrete rule. So the only real question is what determines the name change. I have a few theories:
Only bastards legitimized for the express purpose of inheritance gain the last name. Jon, Ramsay, and Addam and Alyn were all being legitimized because their fathers lacked a male heir. It would be curious then that both the Velayron bastards were legitimized, but I imagine that could be explained by the fact that there was a war on, and Addam's survival wasn't a given by any means.
The Great Bastards were legitimized, but with the understanding that Aegon the Unworthy's firstborn Dareon would be placed on the throne. Given that Dareon had four sons, the chances of any of his half-brothers taking the throne through the line of succession would be remote.
This is my preferred theory, especially since Petyr Baelish strongly implies that even the legitimate Harry Hardyng would become an Arryn if he inherited the Eyrie:
When Robert dies, Harry the Heir becomes Lord Harrold, Defender of the Vale and Lord of the Eyrie. Jon Arryn’s bannermen will never love me, nor our silly, shaking Robert, but they will love their Young Falcon...
Personal preference- Jon Snow always dreamed of becoming a Stark, and Ramsay hated that he was a bastard. It may be that one gets to choose whether or not to take their father's name. The Great Bastards had a pretty awful man for a father, and Brynden in particular seemed to embrace the disdain and fear others had for him. Meanwhile, the other Great Bastards went by names like Blackfyre and Seastar that reflected their backgrounds or personalities.
It may have to do with the mother- All the known Great Bastards were the children of noblewomen. By contrast, Jon (allegedly), Ramsay, and the Velayron boys were the sons of commoners. It might be that when the children of two nobles are legitimized, they don't get a surname or some reason, perhaps to avoid disputes with the noblewoman's husband and children.
A partial answer may be that Brynden was feared and disliked:
Brynden was rumored to be a sinister sorcerer who effectively ruled the kingdom "with spies and spells".
He fell from power and was imprisoned:
It is strongly indicated by Maester Aemon that Brynden was imprisoned in the Red Keep's black cells. The reasons why this imprisonment and the year in which it would have occured are currently unknown.
Later, he was exiled and compelled to join the Night's Watch.
So even if Brynden personally wanted to be known as Targaryen, his many enemies might have called him "Rivers" to emphasise his bastard origins. In a case of history being written by the victors, he became generally known as "Rivers" after his overthrow and exile.
Another consideration is the Targaryen emphasis on purity of lineage, maintained by incestuous marriage. So they might have been particularly hostile to bastards of "impure blood", and denied them the Targaryen name even after they had been legitimized.
Finally, the two examples we have of legitimated bastards taking the "trueborn" family name are
Ramsay Bolton and (at least in his internal monologue) Jon Snow.
Both are northerners, so it is possible this is a northern custom which is not kept in the south -- because of Targaryen ideas about purity, or for other reasons.
