Within Teamwork Chat, you can quickly edit your chat handle that other users will use to chat with you.
Click your profile icon in the top right and select Profile from the dropdown menu.
In your profile settings, you will see a Handle field where you can update the handle text.
By default, the handle that will be chosen for you is your first name however if another user already has a handle with your first name, then yours will default to your first name and first letter of your surname.
For more information, see: Settings within Teamwork Chat