In real life, I don't believe in rebirth. But in the Percy Jackson universe, if Du choose to be reborn, Du will first be immersed in the river Lethe in the underworld, which will leave Du with completely no memories, then the normal process continues.
i think, Du live life again as a different person, but Du still have your personality. Du need to be very kind cause Du could probably be sent to the Fields Of Punishment.
Du become alive again as a new person except with the same soul and personality. Then Du get another chance to achieve Elysium. If Du acieve Elysium three times Du get into the Isles of the Blest.
