Once an edX username has been created at registration it cannot be changed.
You can instead create another account using a different email address to have a different username.
If you would like to use the same email address for the new account then you will need to first associate a different email address to this account.
You can edit your email address from your Account Settings. Click the dropdown arrow next to your username in the upper right corner of any page and select Account.
Delete the old address from the email field, and type in your new email. After you click out of the email text field, you will see the message "We've sent a confirmation message to 'email address'." Click the link in the message to update your email address.
Once updated your original email address will be free to use on the new account.