I have a small VPS with only 512K memory and I added another WordPress site to it. In theory it should have worked OK, there is not a huge amount of traffic on the server and there was enough disk, but I found that after adding the extra WordPress site MySQL would crash each day at around the same time. On doing a lot of research I found the following fix works:
Use the following in [mysqld] section of your /etc/mysql/my.cnf configuration.
performance_schema = off
UPDATE: The above fix seemed to work but after a couple of days MySQL crashed again. The final solution that I found was to upgrade the VPS so that it had 1GB of memory rather than the 512MB that it had.