This article specifically refers to Minecraft: Java Edition usernames 

When signing up for an account and creating a username you may see this error: 

IllegalArgumentException: Registration Failed  

This error occurs when there is already a registered account with the username you have selected. Normally the site will tell you the name is already taken however in some instances this error will show instead. Due to this you will need to select a new username and try again. 

If you are still having issues selecting a unique name, you can contact support here.