It is most probably the system is ran out of inodes. In my case PHP sessions are the issue. Check the inodes by run df -i
Solution :
rm -rf /var/lib/php/sessions/
mkdir /var/lib/php/sessions/ && chmod 1733 /var/lib/php/sessions/
Another reference: http://www.queryadmin.com/1634/bash-script-delete-php-session-files-free-inodes/

By default the /tmp directory is writable by all and files in this directory are usually deleted on reboot. This error could indicate one of the following:

• Somehow permissions on this file or on your /tmp directory were changed and the MySQL process no longer has permission to write the file it needs.
• For some reason the service is being blocked from writing to the file, possible if there is an existing process already running.
• Your disk has run out of space preventing the service from writing the file it needs.

I would recommend first rebooting your droplet. This should clear most if not all of the contents of /tmp and get rid of any hanging processes. If this does not resolve the issue, make sure the permissions and ownership of /tmp is correct.

Hello i’ve got the same issue we can’t connect to the database anymore;

The same problem, my case was out of disk, I rebooted and it worked fine.