Naruto's naming system always confuses me. Naruto is named Uzumaki, after his mother. But Boruto is named Uzumaki, after his father.
So whose surname is taken while naming a child in Narutoverse? Are there some governing factors?
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First, lets start off with the facts.
As for the naming system, they typically get the fathers name. But as I have mentioned, Naruto being named Namikaze would have put him in great danger as Minato had many enemies who sought his life.
Hiruzen(the third Hokage) Decided to put Naruto's Surname as Uzumaki because the Uzumaki was a clan which has Split after the village was destroyed,which meant that Most of the Uzumakis were refugees, which meant that it would be harder to confirm that Naruto was Minato's child. (emphasis mine)
Well, that's quite all there is to it. It wasn't wanted for anybody to know Naruto was Minato's son. This was to protect Naruto. And as I have previously mentioned, the wife will usually take the fathers name and undoubtedly get the last name of the father, such as Himawari and Boruto Uzumaki.
Although it usually depends on the situation at hand, Naruto/Boruto characters will usually get their fathers name, with a few rare cases, as explained above.
Further reading: Uzumaki Clan