What lubricants can be safely added to a laptop fan?

If you are hearing noises, it's already way too late. The bearings are worn and allowing the fan to shift its axis slightly, causing rubbing that wears the bearings more. Adding lubricants can make the problem better for a short period of time by filling the space around the worn parts of the bearings, reducing the fan's ability to shift temporarily. But very soon the lubricant will get squeezed out and the fan axis will shift again. Replace the fan.

One tried and true lubricant is sewing machine oil. Sewing machine oil is designed to work well with parts that move at high speeds, which is exactly what your computer fans are.

What David Schwartz said is mostly correct. By the time a fan is tilted/wearing on bearings, the fan's days are numbered. But if you can catch it quickly, you can prolong the life of the fan for quite a while.

Here's a little tutorial on how to go about it.

It's a bit too long to copy/paste it all over into an SE answer, but here's the gist:

1. The center of your fan has a sticker. Remove the sticker. When removing the sticker, be sure to be extra careful. You want to place the sticker back on the fan later, so it needs to remain intact.
2. Underneath the sticker is a little rubber 'plug'. Under the plug is a little metal cylinder (most of which you can't see). This metal cylinder is made of a porous metal.
3. Drop one and only one drop of oil on the visible part of the cylinder.
4. Put the plug back in place.
5. Put the sticker back in place.
6. After you turn the computer back on and the fan whirrs into life, it might take a little bit of time for the oil to distribute.

Desktop fans are replaced easily enough (they're all pretty cheap), but since you mentioned this is a laptop - which often have unique/hard-to-find components - a replacement might not be feasible. So sewing machine oil might be your best bet.

I had a laptop fan that was noisy. I opened it up and cleaned the axle with some alcohol. I then used a needle dipped with alcohol to clean the slot/shaft/hole/? where the axle goes. There was some dirt in there that caused the axis of the axle to be slightly off. This caused the blades of the fan to graze the case, causing the noise. Then I used some mineral oil on the shaft and used the needle to get some in the hole. It has been running perfectly for 2 years now. FYI, mineral oil is non conductive. Ive fixed several laptop fans with this method.

My recommendation would be NOT to add ANY lubricant. The computer fans are designed to be serviceless. ANd you definitely don't want anything dripping anything onto anything inside a computer... There are several possibilities as to what is the cause:

1. Fan (or fan/heatsink assembly) is not properly screwed in place and what you hear is harmonic resonance.
2. Computer has been opened and you have a cable that hangs over (or sticks out under) the fan and causes the rattle.
3. Fan is dying and this is it's swan song.

My bet would be on last one, though. If it happens on start and stop only it means that probably it got overheated or damaged at some point and it has deformed from temperature. Had this issue in laptop once - later fan started to cut out after some time and finally failed completely. Warranty job, though.

If there is a crucial need to jump-start or ease the fan's working it is much better idea to pull it out completely, give it a good (but CAREFULL!) compressed air cleaning and give it an alcohol soaking bath (isopropyl alcohol is cheap and is nearly as wondrous substance as WD-40). Dry it afterwards and put back in the machine. If the fan was dirty this should do the trick.

EDIT ON LUBRICATION: Any lubricant that would do anything good needs to be sticky (which means viscosity must be high enough), so that it sticks to the fan (duh.), but not too high so it can penetrate to the spot where it's needed. Since fans are somewhat hi-speed, most of the lubricant will get squeezed back out anyway. That means the dust particles and grit have something to stick also. Normally this is happening mostly due to static charge, so it means you can remove any dirt quite easily with airspray. With lubricant you get caking, which can be, after a while, basically only dissolved to get rid of. So WD-40 time (again) or alcohol bath. And after that lubricate again, since WD-40 is not lubricant, but a solvent. Since you end up with most of the lubricant on the outside, and with that speed it will be very fine mist, it will end up inside the heatsink. And thus more caking and more WD-40. That's why I mention viscosity - adequate oils will have low viscosity, which means above process will take hours; with high viscosity oils fan motor will be more strained spinning the fan, causing said motor to overheat... Higher temperature will lower viscosity of the oil, which will cause it to drain faster. And we're in a 'while-then' loop really... There is a reason why in case of high-rpm systems lubrication is applied to sealed components and it's usually grease.

I'm not saying lubrication is a dumb idea (just touching on mildly stupid), but I would see it as an emergency solution only, not normal usage, and with understanding there will be part replacement sooner than later.

The fact that you can doesn't mean you should. Laptop fans are small, high-speed devices. They are designed without any lubrication whatsoever. So that should give you pause right there. But just to drive the point home - since they're designed that way (and, to boot, this is because they are designed as a dust-proof devices, for obvious reasons), only way to meaningfully lubricate a laptop fan you have to tear it down... Which is another "pause and think before you do it" moment: disassemble a part designed not to be disassembled and then put it back together with something extra added.