I'm a newbie, but this might be an easy answer for you:
Remember that you can mount partitions anywhere, not just at / or /mnt.
Suppose your big log files are going in /var/biglogs/ and that directory is on a partition that is too small.
Find a new, big partition somewhere (maybe install a new hdd?), make a filesystem on it and then mount it as /var/biglogs.
Alternatively, you could mount your new partition as /ServerLogs and then tell your server that it should put its logs there, not wherever it is currently putting them (this information will probably be in a config file somewhere).
Hope this helps.