You set your name in the preview screen when you first enter a Gather Space. The name you enter is tied to the Space. This means you can have a different default name for every Space that you enter. You can always change your name on a Space by Space basis. If you have created an account, you can set a default name that will automatically be used when you enter a new space.
Whenever you return to Gather or visit other Gather Spaces, you can always change your name by clicking "Edit Character" which will take you to the dialogue box shown above.
Changing Your Name After Entering Space
If you're already in a Gather space, go to the control panel at the center bottom of your screen and click on your name to open your User Options.
Click "Edit" near the top of the new window that pops up. That will open your User Settings where you have the option to change your name and microphone/webcam/speakers.
Your name will be automatically edited and changed dynamically as you type into this box. You can close the menu once you have entered in your desired name. There is no need to click a save button.