My Profile → Edit Account → Your name. Change it, save it, done.
What changes
Your name as it appears now: in chat, on your profile, and next to a bid in a live show. The change is immediate and applies everywhere the name is read live.
What does not change
Your order history. Past orders keep the name they were placed under. Nothing is rewritten.
Your invoices. An invoice is a tax document for a sale that already happened, and it keeps the name on it at the time. This is correct rather than a limitation — a rewritten invoice is not an invoice.
Your username. That is a separate thing: the @handle in your profile link. Changing your name does not change your link, and if you have not set a username you will see Pick a username first (Edit profile).
If you sell
Your store name is separate from your name, and it is the one buyers see on your shopfront and your listings. Changing your personal name does not touch it, and vice versa.
It is worth knowing which one people actually see where: the store name carries your listings and your shop page; your personal name carries your chat messages and your bids.
In a live show hosted by someone on your team, the store is named with the host alongside — buyers are never misled about who they are talking to.
What a name cannot be
The usual: not someone else's, not a brand you have no right to, not an impersonation of Metzee or its staff, and not something that would get a message removed if you typed it in chat.
A name chosen to look official is the one that gets acted on fastest, because the harm is to other people rather than to taste.
If you want to be less visible
Changing your name does not hide you from a seller you have bought from — they get what they need to deliver your order, which is the address and a contact number.
What a name change does do is separate your bidding in a public room from the name your friends know. That is a legitimate thing to want, and this is the setting for it.
More: can I have two accounts, how to create an account, can I see who I am bidding against.