The easiest way would be to make a new /var that is bigger than your current one.
This should be done offline, so use a live CD so that you can modify the OS. The following assumes
- / (root) is /dev/sda1
- /var is /dev/sda2
- You will create a new /var in /dev/sda3
Boot the livecd, and make some mountpoints:
mkdir /mnt/root
mkdir /mnt/var
mkdir /mnt/newvar
Now use parted or fdisk to create a new partition for var - /dev/sda3
Then make a filesystem in /dev/sda3 for the new var:
mkfs.ext4 /dev/sda3
Mount everything:
mount /dev/sda1 /mnt/root
mount /dev/sda2 /mnt/var
mount /dev/sda3 /mnt/newvar
Now copy everything from the old var to the new var:
rsync -ax --progress /mnt/var/ /mnt/newvar/
Now edit the /mnt/root/etc/fstab file to reflect the new var location:
/dev/sdb3 /var ext4 defaults 0 0
Then reboot.
Note that you will need to modify the devs to match your setup, and make sure the fstab line contains the appropriate filesystem (you do not have to use ext4, just a suggestion)