The first character you create on your account will have a first and last name. All new cloning or characters created from that point will share that unique last name.
You have to choose a unique name that hasn't been chosen by any players previous to you.
"All new characters you create from this point forward will share the same last name. Please verify that this is the last name you wish to use for all your characters."
"Your name must start with a capital letter, and contain only letters and numbers."
"Your name can be no shorter than 3 characters."