Your username is used to create a shareable link to your profile and is an important unique identifier for your account. The username that you choose when you create your account is permanent and cannot be changed.
However, you can change your display name any time. Your display name is what shows in your shop and on all product listings. Changes to your display name can be made from your account page here.
If you must change your username, you will need to delete your account and create a new one.