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