Two things spring to mind:
* Are your logfiles growing larger than you need them to be?
* Where are your databases stored?
Check the (many) log locations and see if there are any files that contain only data about time periods too far in the past for you to need to refer to them. If any such files are present, either just remove them or back them up elsewhere first just in case you do need to refer to them later. Make sure you don't delete log files that contain recent data - you may need them for problem diagnostics and they may be needed by part of cPanel for, for instance, bandwidth accounting and other resource tracking functions.
When I last administrated a cPanel server (which is some time ago, so things may be different now) it didn't see to purge many old logs. The per-account Apache logs were a particular culprit.
This is moot if your log files are not held on the partition that is nearly full, of course. See this graphic for a guide to places to look for log files in.
cPanel's default setup used to keep mySQL databases under /var/lib/mysql/ which could be a problem if this is still the case, you have large databases, and
/var is part of your root filesystem instead of being in its own separately mounted filesystem.
If this is the case then you could move these data files elsewhere, such as a subdirectory of
/home or where-ever you have enough space, either by moving them and reconfiguring mySQL to look at the new place, or moving then and making /var/lib/mysql/ a symlink to the new place (remember to stop mySQL before making changes). Using the symlink option may be safer than reconfiguring mySQL as there is less chance of a later cPanel update accidentally reverting your change.
The same goes for other database providers too, i.e. if you use postgres its data files may also be in an inconvenient place.