MySQL data is stored in `/var/lib`. There is no more space in `/var`. MySQL does not start. Pretty simple actually. Have a look at `/var/log` and clean it up. I would recommend something like (delete all `gz` files in `/var/log`) :

 $ find /var/log -iname "*.gz" -delete

Of course, you might wanna check what's being deleted first:

 $ find /var/log -iname "*.gz"

1.7TB is given to `/home`. However `/var` is not `/home`. So `/home` is not being used. Simple as well.

You clearly should have separated `/var` on a much larger partition when you set up your system. Now, the best solution I can come accross is moving the MySQL datadir:

 $ mv /var/lib/mysql /home
 $ ln -s /home/mysql /var/lib/mysql

If you don't want to use a link, you should reconfigure MySQL to use the new datadir stored in `/home`. Edit `/etc/mysql/my.cnf` and change this...

 datadir = /var/lib/mysql

to this:

 datadir = /home/mysql

Now that MySQL has space available, it should start without trouble. You should really reconsider your partition scheme if you're hosting such an important database though.