There are quite a few names that mean Phoenix or relate to fire or other topics related to Phoenixes. These names include Bedelia, Bren, Helia, Missja, Aiden, Azar, Col, Fintan, Kai, Ash, and Flint.
As for names that relate to rebirth, rejuvenation, and new beginnings, there are also plenty of names that fit that bill. These names include Roxanna, Renata, Neoma, Kady, Winona, Alba, Amaryllis, Anastasia, Aurora, Avril, Chloe, Dawn, Eos, Abel, Irvin, Janus, Neo, Kit, Tan, Renee, and Nova.