According to our documentation the MySQL Connector must be downloaded and moved to <BAMBOO_INSTALL>/atlassian-bamboo/WEB-INF/lib/.

This is not concise among Atlassian products. For example, in FishEye the connector must be moved to <FISHEYE_INST>/lib.
In Bamboo, the driver should be in an equivalent directory, such as <BAMBOO_HOME>/lib.

The advantage of doing this is that users won't forget to download and move the MySQL connector file every time they upgrade Bamboo.