My vote is NO OIL!
A couple decades ago you could disassemble a fan to oil ONLY the bearings lightly and then re-assemble cleanly for a year or more of extra lifetime. Today there is no way to open it up without breaking it, and then no way to re-assemble properly. If you just a little oil on the outside it is NOT going to reach the bearings. If you put on a whole lot to be sure, the oil gets everywhere inside the fan and causes other problems.
OP, you diagnosed unbalanced fan, and others have suggested wearing bearings - a different cause. To help diagnose, here is the typical sequence of bearing wear on fans 3 to 6 years old. The basic mechanism is that the outer sleeve wears away to a slightly too-large hole, allowing the shaft to "rattle" in the loose fit.
1. As the system heats up in normal use, the shaft expands its diameter just a little more than the sleeve expands its hole. So on a COLD start-up after several hours off, the fan makes this noise for 15 sec to a minute, then the noise stops. If you reboot the noise stays away. If you shut down long enough to cool the fan off, the noise comes back on re-start for a short time.
2. Over the space of a few months the period of start-up noise gets longer.
3. Eventually the noise never stops. This is the last stage of life for this fan and it probably is running too slow to do its full job. It must be replaced.
4. If not replaced, ultimately the fan bearings seize up and it will NOT run at all, so you get no cooling from it.