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