Ubuntu 12.04, Nginx and MySQL issue with WordPress Database

512MB should be fine, especially if you just started building your site.

I’ve ran single WP installs on DigitalOcean droplets with 512MB of RAM without any issues in the past, also on NginX. Obviously, server stability also greatly depends on how NginX, MySQL etc have been configured (ie. not to consume too much memory and such)

Have you installed any security-related plugins? On what DO zone are you? From personal experience, they are having issues with some hypervisors on the Amsterdam zone currently.

If you are somewhat inexperienced related to sysadmin-tasks, try downloading an SFTP client such as BitVise Tunnelier to login to your droplet, visually browse through folders and grab/download log files on your computer for further analysis.

Assuming you have not modified the default log locations, you should be able to find relevant logs in or at: /var/log/nginx, /var/log/mysql, /var/log/mysql.err, /var/log/mysql.log

Edit:
I found this for you:
https://groups.drupal.org/node/218439

Disregard the fact that it appears to be Drupal related, the problem you’re mentioning might be related to a faulty php-fpm config.

Another possibility I think might be incorrect Nginx permalink rewrite rules.

Are you able to paste here/upload on pastie.org your NginX config file(s) ? Feel free to redact your IP/domain name.