Password is 'password' by default.

as SSH message says:

WELCOME TO DIGITALOCEAN'S [..]
-------------------

Please be sure to change your root MySQL password;
current MySQL password is "password".

You can change the MySQL password with the following command:

mysqladmin -u root -p'password' password newpassword

You may try

cat /root/.my.cnf

Usually, you should run mysql_secure_installation after installation is complete to set the root password (because there isn't one) and secure the installation.

On the LAMP one click application, when you login to the droplet the MySQL root password will be displayed in the terminal. It's part of the MOTD, or message of the day. To display the message again, run:

cat /etc/motd.tail

@alexandre: That's no longer true. As part of upgrading the LAMP app to be based on Ubuntu 14.04, now a random password is generated for each droplet when it is spun up.