You may also want to try a different theme…maybe just one of the default WordPress themes. Sometimes developers will write some hefty functions into themes that can cause problems.
I run several different droplets that are 512mb, WordPress sites on each of them with no problems. I did have to hunt down an issue on one of the droplets, but it turned out to be a wp-cron.
Also, there are couple of other things to check:
How much traffic does your site get? Do you have analytics installed so you can look at that?
Do you do anything to guard against brute-force attacks? If not, you should at least use login-lockdown plugin, which can help. It will also log any false attempts to login.
If you install that plugin, and then look in the settings an hour later and see tons of login attempts, then that could be a problem (but there are solutions..)
Also, do you have any other applications installed like phpMyadmin? Just to get a full picture of what is going on.
And finally, here is a script that will restart MySQL for you if it goes down:
that will not solve your problem, but it will at least keep your website up while you trouble-shoot.