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: 
A material that doesn't check most of your requirements but could still fit your use case is clean, liquid water:

Water in itself is in abundance in the universe, however not all forms of water are useful for living life.

This material can be synthesized and we can extract clean water from ice which is abundant. However, for a civilization, extracting or synthesizing water for the entire population and on such a massive scale must be incredibly expensive. We're talking about potentially terraforming entire planets for the investment to be worth it. On the other hand, finding a planet that already has large volumes of liquid water is much much more valuable and cheaper to use.

Technically your civilization cannot survive without water. However as mentioned earlier it is in abundance in the universe. What is valuable is the scale and access of liquid water that your civilization has. So a ship crashing in a planet with large ice caps may extract enough water to survive. However your civilization will need huge investments to install a settlement there.

So civilizations will go to war to create their own settlements on planets with liquid water first before getting the ones with ice caps which are more expensive to use.
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.