Rebirth is the process of being born into a new body after death. In the Underworld, one can be reborn by first getting dipped in the River Lethe to erase their memories of their previous life and then being born again in the mortal world. If the soul reaches Elysium three times, they can enter the Isles of the Blest.
After sacrificing himself to scatter Kronos, Luke Castellan lays dying on the floor of throne room of Olympus. When a saddened Annabeth Chase says he would surely achieve Elysium for his act, the son of Hermes mutters: "Think rebirth...Try for three times...Isles of the Blest...."
On the Quest for Thanatos, Frank Zhang and Hazel Levesque shared a blackout. Hazel went to the Underworld with Frank, explaining to him about what happened after she died. In the Fields of Asphodel, Nico di Angelo came to try to find his sister Bianca so that he could bring her soul out of the Underworld but could not find her, Nico then found Hazel and brought Hazel instead. He explained to Hazel that Bianca had been reborn and that he would never see her again.
Nico di Angelo isn't sure if Jason Grace is in Elysium or if he has possibly chosen rebirth. When he looks to Apollo for answers, the god admits that he doesn't know either as it isn't his jurisdiction.
|Not Part of Riordanverse|
Riley Oh mentions that the Gifted Community believes that when one dies and goes to the spiritrealm, they can choose to either stay in the afterlife or have their memories erased and be reborn as someone close to their loved ones in their previous life.