Hands are the only character I have here and can think of that actively chose their own name. They chose Hands because it was the first thing they saw after they gained sentience as a warforged, also because they like to work with their hands and invent stuff (they work as a decorator in a travelling fae circus now)
Though a lot of my characters change names in their lore, but most of them get re-named by the people they like/trust and choose to keep it, which is an interesting tendency that I only noticed now!
Button specifically responds only to a nickname that his new caretaker gave him.
Fungi (and their travelling partner Shroom) had some bad associations with their old names, given to them by their villages that only reflected their "function", so they chose to keep the nicknames that their new overworld friend gave them as their names.
Cicada is a forest gnome, and they traditionally have a lot of names, as they like to give them to each other and keep them. Her full name at the moment is Brina Cicada Flea Timberson, Brina being the name given by her mother, Flea - nickname her grandma gave her (because she loves insects, has red hair and always on the move), Cicada - name of her own choosing (her favourite insect) and Timberson being her clan's name.
Carl only remembered the first letter of his actual name, so his new friend came up with a new name that also started with C and he ran with it
I also have two characters from my writing that I did not post here, but they changed their names a lot so it's a good example:
Auka initially had a normal human name, but it became alien to her when she started to turn into a werewolf, so she choose Other > The Moon called her Little Wolf > the Spring Foxes called her Pumpkin Daughter of the Moon (Pum'da'moo) and she was responding to all of those, but in the end she found herself a girlfriend and asked her to come up with a name to call her. She chose Auka (or Howly in english) because it reminded her of the wolves, howling sound and childhood and it sounded soft.
Auka in her turn also came up with a name for her gf as a tradition - she called her Juniper (Mozjevelnitca) because she smelled like juniper and pine trees