I've never been sure about this thing here. When a Guardian comes back in life, he remembers nothing about his own past. Always found this very intriguing but then... does a Guardian chose his own name after rebirth? Or does he remember it? I know that many Guardians have dreams about their past so, maybe, some of them could dream scenes of their past and ri-discover their names, but what for the others? And what about the Exos? Is the same thing for them?
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Well seeing as there are no Guardians named Terry, Mohamed, Alison, or Pedro, or any other common name, I'd wager that they might remember some aspect of their birth given name, but they pick their own name later on. Unless culture and names have changed drastically in the relatively short 300 year span of the golden age. Seriously, who would name their kid Shaxx? Or Felwinter?