/var/lib/mysql is mounted on partition / which is now full. Is there a way to mount this under /home/mysql for more disk space?
It could be implemented like this (WARNING: not tested)
Always have a backup before you start moving the data directory.
Ideally, you'd give it it's own mountpoint on it's own partition, which can be mounted to
Failing that, you do have the option of using a loopback filesystem, though it definitely does introduce some I/O delays that MySQL may notice. Only you can know if increasing I/O latencies a bit will harm your expected performance.
The rough procedure....
At that point, you should have a larger /var/lib/mysql actually hosted from your larger partition.