I'm looking to know about minerals that space miners can extract from other planet surfaces or directly from asteroids. Something that Earth as a planet may need in the future.
This answer stands as an historical artifact, obsolete because of an edit to the question's tags.
Energy-rich minerals of unknown composition from the surface of Mercury.
Mercury, the planet has an eccentric orbit leading it to be at it's closest point to the sun it is bathed in searingly hot unfiltered solar radiation and cosmic rays. Mercury turns slowly, meaning that in each day the surface absorbs prodigious amounts of solar radiation relentlessly.
This effect has a startling consequence. The surface minerals, when a sample finally made it back to Earth, - a small fraction of them show an astonishing property - if they are subjected to a strong magnetic field then they re-emit that Energy to which they have been subject for millions of years in the form of gravitational force. The unfortunate Lab assistant who discovered this effect, has received a state of the art prosthetic foot.
On discovery of this property, of course it was investigated further, and found to be controllable, it now represents the biggest potential for space exploration and cheap and clean in-atmosphere flight, and of course cheap energy. - Dangerous to venture to Mercury to fetch in any quantity - this mineral, particularly when the stakes and the demand are so high, the governments and corporations scrabble for control of this fabulous resource, but it attracts a certain "frontiersman" type with nothing to lose and everything to get away from. Over to the writer.
Earth is running dangerously low on helium.
It's important enough that America has a strategic helium reserve of 1 billion cubic meters, although they may be selling it off.
The LZ129 Hindenburg zeppelin was designed to fly with helium, but the USA decided not to sell the necessary amounts required to Nazi Germany.
Earth is made from the same raw materials as asteroids and comets. In fact, the current belief is that the Earth formed by accretion of a piece of the solar nebula, which is where everything else in the solar system comes from.
Now, some of the heavier elements like iron may have sunk in large quantities to the core as the Earth cooled after its formation. But asteroids falling on Earth also deposited a lot of materials on its surface. Meteoric iron is a thing.
If you are going into space for raw materials, you are just getting those materials in a more expensive way.
The only things outside Earth that do not exist on or in Earth are exotic materials that are only formed in extreme conditions. For example, some models predict that Jupiter's core is made of metallic hydrogen. But that is not something you can mine from an asteroid surface.
Note that if you had asked what people might need when far from Earth, then asteroid mining might become economically feasible.
There's not a huge amount of utility in shipping stuff found in space back down to Earth's surface... after all, we have an awful lot of stuff here and there are plenty of places we haven't yet tried to exploit. Exceptions might be made for materials that are limited in availability and might be largely found in particularly picturesque regions of the globe, or in particularly politically unstable countries, but even then it'd have to be pretty bad to justify going into space to get it. There may be some exceptions... siderophile elements are likely to be common in the earth's core (due to their density) and less common at the surface, but they'll be a lot more accessible in asteroids. Some quite rich people are very interested in the possibilities of mining huge quantities of precious metals but, y'know, there's only so much platinum the world needs and one big find could crash the market.
Some people might say "Helium 3!" which is useful, but the only places you'll find it in serious quantity are gas-giant atmospheres, and you'll need to spend so much energy getting the damn stuff back out again that there'll be little benefit to using it as fuel afterwards.
The benefit in asteroid mining is that the resources you are harvesting are very easily shippable around the solar system, and don't need to be dragged out of a huge, deep gravity well. If you also had refineries in space then it would be vastly cheaper to supply them from space-based sources. It is possible that there are some useful refined materials that are best made under zero gravity and vacuum conditions (foamed metals, maybe?) but it isn't obvious that they'd be useful enough to justify such expensive to set-up and inconvenient to operate industry.
Space-based fabrication might make sense for things that need to be in space... if you can make satellites in space, it'd certainly be better than hauling them up from ground level, but the cost of establishing that industry is going to be very high, and return on investment will take a long time. Some people are more optimistic about the possibilities here, so perhaps it'll be easier than I suspect.
What remains, then, is probably the most boring: ice, and rock. Ice to fuel rockets (because hauling fuel in bulk out of Earth's gravity well is a right pain) and rock to build with (because it makes for nice radiation and micrometeorite shielding) should you wish to put more people and delicate equipment in space.
Minerals? Not much, until the transport costs come down.
Heavy metals (Gold, platinum, irridium...) are more common in certain classes of asteroids. However useful these metals are, their high value on earth is due to scarcity. All the recovered gold in the world would fit in something like a 65 foot cube. An asteroid with a trillion tonnes of gold would drop the price to the point that gold was used for wiring houses. (Probably not: Not strong enough)
However some processes may well work better in space -- microgravity may allow very large single matrix crystals -- a process that is difficult here. Any process that needs a good vacuum can do well in space. Getting stuff down from orbit is a lot cheaper (but still not cheap) than getting it up. Probably worth it for anything with a street value meansured in a few thousand dollars per kilogram. (Computer chips, drugs, vaccines,...)
Hal Clement in one of his short stories speculates that in the cold of the outer planets you may have compounds that just aren't stable at normal temps. Whether there are any that are abundant enough to become 'geology' is up in the air.