Your username
Your username is the name at the end of your link. It is also the one part of Byze that is public: anyone can look it up and see the keys it points at.
You pick it once, when you make your account, and it stays with you. If you want more names, each extra one is a separate purchase and costs 0.1 SOL. That is not open yet, so for now your account comes with the one name you chose.
Names are taken by whoever gets there first, so short common words go quickly. Some are held back for the service itself, and some are refused because they would read as us speaking.