What resources would be most valuable to enhancing space travel, establishing new colonies, be cause for invasion, etc?
All of Them
Spaceflight occurs at the end of a large, interconnected web of technological development.
You need chemistry, refrigeration, pumps, metallurgy, electronics, computers, precision manufacturing, structural engineering, medicine - with math underneath all of those.
You also need 'soft' technologies like project management.
For any kind of long-distance space travel, like colonizing other bodies, you need biology and even more chemistry.
The resources needed for spaceflight include ~30 elements on the periodic table, all sorts of compounds built up from those elements, and the knowledge and skills to turn those compounds into useful (and often very complicated) objects.
There's no one thread you can pull out of that tapestry and call 'the most important' or 'the biggest target for invasion'.
It's hard to say without knowing what technology the colonising societies are using. Prior to the 1930s, Uranium's primary use was as a yellow pigment. Nobody could have guessed how valuable it would become!
It's also important to remember that there's almost no material substance that could be gained from invading another star system that couldn't be acquired more easily from asteroids, moons, or other celestial bodies. If you can travel between the stars, you can meet your material needs quite easily.
So for the moment, let's put aside the notion of conventional resources; they're just not rare enough to drive conflict or colonisation. There are two other things that might.
We're already starting to see something of this in the modern world. The limiting factor in a lot of situations isn't the availability of raw materials, but access to the information and knowledge needed to take advantage of those materials. A situation where wars are fought over the control of major data archives or universities could be imagined.
This is the one thing we've discovered that seems to be genuinely rare - indeed, unique to Earth, so far. The complex chemicals produced by living things, indeed the complex process of life itself, seems to be rare, at least in our solar system, and so are environments that can naturally support life. A war could be fought over access to life-bearing planets, or perhaps planets where a particular form of life can grow (echoes here of Dune and the Spice that can only be found on Arrakis).
If you want both information and complex chemistry, and a base of people ready to use that knowledge, then a large population with a developed economy would be a useful thing to have. Wars might be fought over the right to exploit a given population, with whole worlds changing hands. An interesting wrinkle would be that all parties would want to keep the fighting away from the planets, to avoid damage to the prize.
It really depends on what technologies are in use... but when it comes down to it, the most valuable commodity could very well be certain forms of biomatter and other organic chemicals.
Why? Most metals and resources needed for building structures can already befound in space, eg. on asteroids, moons and barren planets. The same may also apply with fuel (radioactive isotopes or especially hydrogen). However, life, especially life that is usable for human consumption, is still bound to be relatively rare.
Life requires a set of strict parameters to even evolve, moreso to maintain. Even then, if encountered on another planet, it might not be edible or usable by us. An alien tree or crop might as well contain dangerous chemicals, or may have life processes completely different from what is found on Earth.
Any interstellar expedition or colonization would most likely need something to build and maintain a friendly ecosystem, or at least be able to provide suitable food in the long run.
Examples of valuable biomatter could be:
- Plants and seeds, which can help sustain a livable oxygen atmosphere and supply food for humans in spaceships, on stations and in colonies.
- Fungi may also enable a more varied food supply, but are also vital for maintaining a stable ecosystem, primarily in their role of "decomposers" of "unusable" biomatter. Some fungi may also supply medicine (penicillin being a good example). Yeasts may also help extend the longevity or shelf life of various foods, or even help in creating a safe water supply, for which they were used until fairly recently (beer/wine).
- Various virii and bacteria, which not only can assist in controlled fermentation of food and oxygen supply, but may also be integral components in medicines and vaccines. They are also important in maintaining an ecological cycle.
- Certain animals, alien or not, may have properties or may supply resources that are otherwise useful. Their meat and byproducts may supply vital proteins and aminoacids that a fully vegan diet can never sustain
Also the lack of "friendly" biomatter may be a perfect motivation for invasion.The habitat ecosystem on colony A may have failed - the only way to get the needed biomatter may likely be by trade - or barring other options - invasion...
Other invaluable resources may include resources that are otherwise unique or monopolized. Examples of these may be "spice" from the Dune universe or Quirium from the Elite games, which both play a role for being nescessary for FTL or hyperspace travel.
While Dune's spice is a natural resource, the cultivation of it in space is impractial, mainly because of the sheer size of the Arrakian sand worms that produce the commodity.
On the other hand, Quirium from the Elite universe was a manufactured commodity - however the resource was monopolized by a company called "The Galactic Cooperative" or "GalCop", who closely guarded the formula for the fuel, much like Coca Cola guards the recipe of their soft drink. Within the lore of the continuity, GalCop was dissolved around 150 years before the latest game, and humanity was relegated to using slower and more inefficient methods of travelling the galaxy, until better alternatives were invented (the Frame Shift Drive).
Worlds or other places that are essential to producing important or priceless resources such as these, are almost always very obvious targets for attack or invasion.