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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Personally, it makes more sense to me if they are able to remember their old name. When they are reborn, they know how to walk, and talk, they don't relearn these things. They already know. It is probably the same with a name; it is your identity, so it persists through death. Unless we forget who we are each time we die.