/dev/mapper/VolGroup-lv_root has no more space left?

The easiest and fastest option is to create a directory in /home and move all data of /var to that directory and create soft link of /var to /home/var. The steps will be as follows.

1. mkdir /home/var

2. stop all the services. For example /etc/init.d/mysqld stop

3. cp -pr /var/* /home/var

4. rm -rf /var

5. ln -s /home/var /var

6. start the services again.

Note: May you can only move mysql data which is on /var/lib/mysql and create softlink to that.

You can :

1. Add hard disk

2. Move mysql DB files to other FS (in your case you have just /home FS left )

3. You can resize your LVM volume to make /home smaller and / bigger (be sure to have backup of course :) )

This option is most straight forward, but LVM strongly recommend to backup your data before resizing

http://www.ibm.com/developerworks/linux/library/l-resizing-partitions-2/index.html

UPDATE (based on comment)

if I add hard disk what must I adjust on my mysql setting to expand the data to the hard disk? If option to mean I have to move mysql over what steps need to be taken for that?

Quite easy:

1. Stop mysql
2. Open /etc/my.cnf and find datadir= - here are your mysql datafiles
3. Copy/Move them over to your new desired location
4. Update datadir= with your new path
5. Start Mysql
6. Take a beer or whatever you want, cause you are done

How tough to expand the /root?

Read the link above which I already posted.Step by step guide is included there with comprehensive explanation.

Normally what the home partition is to store? Any important files?

It contain user files and setting.Eg your desktop etc.

For system healt /home is not important and can be remounted anytime.

I would suggest, as the first first, to check the size of /var/log. Possibly archive older logs. They can clog up you root partition without you noticing, and most older logs might not be necessary.