No; 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, you will first need to associate a different email address to your existing account.
You can edit your email address from your Account Settings. Select the dropdown arrow next to your username in the upper right corner of any page and select Account.
Remove the current 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 no longer be associated with your current edX account, and will be free to use on the new account with your new username.