If you have a ext3 volume on LVM you can easily change this volume.
When you want to extend a volume you can do this with lvextend.
To extend a volume with 10GB do the following:

# lvextend -L+10G /dev/<volgroup>/<volume>

Extending logical volume <volume> to 14.88 GB Logical volume <volume> successfully resized.

After this you have to extend/resize the filesystem:

# resize2fs /dev// resize2fs 1.39 (29-May-2006) Filesystem at /dev/VolGroup00/os is mounted on /; on-line resizing required Performing an on-line resize of /dev/VolGroup00/os to 3899392 (4k) blocks. The filesystem on /dev/VolGroup00/os is now 3899392 blocks long.

You can do this without any problems on a live root file system.