What exactly happens when you are reborn?

In real life, I don't believe in rebirth. But in the Percy Jackson universe, if you choose to be reborn, you will first be immersed in the river Lethe in the underworld, which will leave you with completely no memories, then the normal process continues.

i think, you live life again as a different person, but you still have your personality. you need to be very kind cause you could probably be sent to the Fields Of Punishment.

You become alive again as a new person except with the same soul and personality. Then you get another chance to achieve Elysium. If you acieve Elysium three times you get into the Isles of the Blest.