If you wish to use the root MySQL user with PHPMyAdmin you will need to set a password for that user. The Wordpress One-Click uses a different user account for Wordpress itself and you could choose to connect using this already configured user to access your WP database. This information should be found in your wp-config.php in your Wordpress installation directory.

Thanks for your help. I was close to this answer but the exact commands to execute to complete this step eluded me, being that I’m not a commandline expert! Your link was helpful but I ended up using this simpler command on a fresh droplet:

$ mysqladmin -u root password NEWPASSWORDHERE

After this the installation proceeded without a hitch.

For the sake of any other users who are also having issues with this process, the outcome of this is three users with access to phpMyAdmin:

1. root - with password you set with the above command. Access to all DBs incl phpmyadmin and wordpress
2. phpmyadmin - with password set during phpMyAdmin installation. Seems to only have access to the phpMyAdmin DB
3. wordpress - with password found in wp-config.php setup as part of the One-Click install script. Access only to the WP DB

Thanks again for steering me in the right direction!