"/" filesystem 100% filled

I have a problem with my QNAP (Intel 64bit, 8GB RAM). System software is some bizarre mutation of Ubuntu. For example, filesystem "/" has a capacity of only 290 MB and its normal occupation is 90%. I received such information from QNAP help-desk. The rest of the filesystems (like /var/log, /root, /lib etc. etc) are mounted to "/". Except - surprisingly - /etc. I wrote some of my own scripts and something fills my filesystem "/" to 100%. Does any of colleagues have an idea for some script in the shell (QNAP has no bash, only sh), which, say, every 5 minutes from cron, examines which file on the filesystem has increased its size by, say, 100 Kb (compared to the previous test)? If I detected which file is growing, I would move it elsewhere, e.g. via a symbolic link. Do you have any idea from your colleagues?
Please consider that the total capacity of this filesystem is - as I wrote - only 290 MB. So not much. Hence, such testing of a single file system will not burden the NAS too much.