Which distro is it? If it is a newer Red Hat or Fedora Core, there is a graphical program /usr/sbin/system-config-lvm where you can add the free space to the logical volume. Or you can do it from the command line. ( See "man lvm" ). For SuSE, there is a Partitioner program in YaST2 that you can use. Other distros may have a similar program.
If you have partitions mounted traditionally, you could create a partition in the free space and then move contents of the largest directory there. For a workstation, /home is usually the largest. Installing new software, it usually gets installed in /usr/ somewhere. The /usr directory will grow in time and is the normally the second largest directory, so it would make a good candidate as well.
For a server, you may have a small /home directory. Moving /usr or /var might make a better choice.