Question: I have been world-building a setting where a species of advanced aliens invades Earth, only to be forced off after the humans resort to guerilla tactics. In this setting, biological life is abundant in the universe, so Earth is not unique in that regard. The aliens are advanced, but not to the point where they have outgrown the need for raw materials. In the story I have now, the reason that the aliens have not wiped out Earth yet is because using nuclear/chemical/biological weapons of mass destruction might make the earth uninhabitable for a future harvesting operation. I have not determined what resource would best fit this role. However, I have discerned that it has the following properties:

* It is inorganic
* It actually exists in real life
* It is in reasonable abundance on earth, but is rare throughout the
universe has a whole
* The material cannot be reasonably synthesized under all known laws of
physics
* It is valuable enough for an advanced civilization to fight a war
over it, but not so valuable that the civilization cannot survive without it.

Which material, if any, would best fit the above criteria?

Answer: 
Reverse the decision, and consider that where the aliens come from they have a deficiency of a certain element. Though take into account that it would be far easier to mine it from a closer solar system than ours or even ignore us on earth and mine all they would need from Mars or the asteroid belt.

You may perhaps want to reconsider the non-organic exclusion. Since the universe we see is pretty much distributed well from the element producing supernova factories. this dust cloud has more or less the same ratios of that dust cloud. All solar systems are born from dust clouds.  

A bad 90's movie with one of the better reasons for coming to earth was ["I come in peace"](https://en.wikipedia.org/wiki/I_Come_in_Peace). The alien was a drug runner, harvesting endorphins. Point being, you don't want to kill off the drug producing inhabitants, and it's far more plausible that organic chemistry has produced something unique on earth. also drugs are bad, ie: not necessary though valuable.  

PS: Even a low Kardashev scale II civilization is still incredibly more advanced than we are. We would still have no hope of defending ourselves against them.
A huge fraction of the universe is composed of just hydrogen and helium. Something like 98%

Of the remaining elements, they are produced in supernova or for a few the lighter ones, as the end stage of a star running out of fuel.

See: <https://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements>

Of the elements it looks like Ta is the least common.

You can go a step further in rarity by going for particular isotopes.

But remember that any element can, with enough energy be made. Doing this efficiently is left to the engineers.

With made materials you have some additional options. Right now probably the most expensive material per pound are semi-conductors. The cute epoxy box is much larger than the actual chip.

You can go for exotic properties too. High temperature super conductors. (We don't know how to make them yet.) Materials with very large electric charges baked in (electrets) Magnetic monopoles (Do they exist?) Perfect crystals.

George Gamow working with an an early theory of nuclear stability found a theoretical nucleus with a mass of several thousand, shaped as a donut. I don't think the theory is current considered credible. But consider if you had a ring shaped nucleus. Now make a chain of these with the rings linked. You have a material that is stronger by the ration of the strong nuclear force to chemical bonds.

Once you can make nuclear thread, you can either weave it to make nuclear cloth, or link to make nuclear plate.

It would have interesting properties.