Has Corellan named his lightsabers?
What a great question!
Short answer: Not usually… with one exception. The two cerulean blue lightsabers he started using after Tython after he still (normally) uses to the "present day" (post-KOTET) didn't require names for him to remember what they were for. One was the former saber of Laotah, the Selkath jedi knight he encountered on Tython who was already dying of wounds suffered from the Flesh Raiders. Corellan repaired it with a blue crystal at the Forge, and he uses it to this day. That saber is a reminder of his limitations and his failures. His other blue saber was built with the materials Orgun Din gave him, also with a crystal from the cave on Tython. That saber is a reminder of his successes. That perseverance can pay off and allow him to do extraordinary things. Those are the unnamed lightsabers he "normally" uses to this day. But there is an exception.
This is Throne-Breaker. The lightsaber he forged on Odessen with the help of Satele and Darth Marr. Whereas his blue lightsabers are ultimately light-side, a reflection of his core values, Throne-Breaker is balanced. An acknowledgement that there is darkness in himself, and there always will be. He used Throne-Breaker when he fought Arcann on his flagship. He used it when he struck down Vaylin on Odessen. He used it to destroy Valkorion / Vitiate / Tenebrae. He doesn't use it always. In fact, since the fall of the Eternal Empire, he almost never uses it. He now knows that people are not meant to be balanced. Not for very long. And people who try to do it… it doesn't work out. Look at Revan, for example. But Corellan keeps Throne-Breaker carefully hidden in the safest place he knows, knowing that it will be there if he ever needs it again. (No, I am not telling you where he keeps it. )
Thank you so much for the ask!