Can't access droplet after a reboot, can't change root password after a reset

I did a reboot on my droplet, and it became unresponsive via SSH. That was strange and I decided to investigate it via the console. I couldn’t find my original root password, so I reset the root password. The next time I opened the console it asked me to enter a new password, which I did (and yes, I entered the original password first). After that the console window blinked and everything went back to square one: it accepted the original password and asked me to change it immediately. What went wrong, and what can I do about it?