In my science-based novel, Moon colonists need to return to Earth to collect some rare or absent mineral(s) Ideally, something they could derive from seawater. Which minerals might they be...?
The composition of Moon rocks is pretty much the same as that of Earth. However, none of the processes that concentrate specific minerals in one location (i.e. into a mineral deposit useful for mining) exist on the Moon. This is due to the lack of tectonic activity. Therefore in the Moon you would expect the rocks to be much more homogeneous than on Earth. Only maybe meteorites, their impact craters, and ancient volcanic activity (as pointed by @userLTK) might have provided a way to concentrate some minerals in various places. So, due to this homogeneity, to get one gram of let's say copper, you would have to process hundreds of times more rock than would be needed in an Earth-based copper mine. Also as Fred mentioned, there would be many challenges for mining in the Moon, but in the context of your plot those might have been solved. Having enough energy, you can probably get pretty much any mineral from the Moon (for example a mass spectrometer would be an effective but highly inefficient way to mine all the elements from within a piece of rock).
If your plot doesn't require that it be a mineral, I think what they might need the most is some sort of organic compound. Even with current technology there are many chemical compounds that can't be synthesized, and we rely on plants or animals to obtain them. A famous example is Maitotoxin, a small molecule that chemists have tried to synthesize for 22 years without success.
So perhaps there could be a particular organic compound that has been recycled and reused over time in the bodies of the plants and animals on the lunar colony, but over time that compound degrades (by cosmic rays, UV light, or whatever), so animals and/or plants start developing problems due to the lack of this compound that none of the organisms placeable in the colony are capable of synthesizing. Therefore, they need to go back to Earth to get it from some ocean-dwelling creature.
Perhaps it would be something like a vitamin required by plants. Or maybe a compound that could be needed to cure a disease.
It is unlikely anything other than salt would be derived from sea water.
The largest deposit of gold is dissolved within the worlds oceans - in terms of tonnes of metal. The problem with extracting the gold from sea water is the concentration is so low it is uneconomic to do so. It's why we keep mining hard rock sources of most metals. This problem exists for other metals dissolved in the worlds oceans. We are only just starting to mine minerals from the sea floor - see Nautilus Minerals operations off the coast of Papua New Guinea.
In terms of Moon colonists requiring minerals from Earth, various salts or carbonates might be candidates. IF the colonists will be making steel, as opposed to just making iron, on the Moon they would need carbon, which would have to be sourced from Earth.
Allied with the extremely low concentration of metals in sea water, the concentration of minerals/metals in some deposits on the Moon may be very low and it might be cheaper and/or easier to obtain the minerals/metals from Earth.
The other problem that might arise is that the mineral assemblages on the Moon may be different to that on Earth and while mining the minerals might be straight forward extracting the required/useful metals via metallurgical processes may be so problematic, or energy intensive, that it is easier & cheaper to obtain some minerals/metals from Earth.
Unlike on Earth, water in large quantities will not be able to be used for metallurgical processes, such as froth flotation. Similarly, processes requiring the use of large quantities of oxygen will not be able to be used. New metallurgical processes may need to be developed to exploit lunar minerals.
The problem for Moon colonists may not be the lack of minerals, but the lack of other types of resources to assist with metallurgical recovery or they may not be able to have metallurgical processes to exploit minerals on the Moon.
Which Earth minerals would Moon colonists miss the most?
tl;dr All of them.
As mentioned in the other answers here, it's about economics. Extracting an element from a mineral requires energy, chemical processes, equipment, waste management, etc. This has to be done per amount of rock. The more of the element in interest you have in your rock, the more economic it is.
Earth has active geology: hydrothermal processes, plate tectonics, magmatism, surface erosion and weathering. The combination of that causes some element to be concentrated in ore deposits, where the enrichment factor over the average composition can get to several orders of magnitude. That's why we mine gold from lodes and not from your average soil or rock. This is why we mine iron from BIFs and not from sea sand.
None of this happens on the moon. Very few elements are enriched to economic concentrations, maybe some copper and zinc close to ancient volcanic vents. But then, you need to dig the stuff. You need to chemically extract it. You need to do something with the mining waste.
Most likely, you will need more of an element than you can extract. To break some rocks you will need some vanadium or titanium tools, or maybe tungsten carbide. There is wear and tear on the tools. You will need to make new tools. For that you will need to get even more vanadium and tungsten, that are simply not available. Maybe extraction processes require organic compounds. There are no suitable organic compounds on the moon. You will end up having to ship the stuff from Earth just to be able to support a mining operating, that will not produce enough materials to support the mining operation. This would be like dumping money in the rubbish bin.
So, no. Do not do it. Mining on the moon will not be a self sufficient operation.
Feels like a space-ex question and I think a fair bit is likely to be missed in my post, but it's a fun question, so:
The biggest problem with the Moon is no atmosphere. Other problems are no protection from UV light, protection from cosmic rays and very fine, small and jagged lunar dust. Annoying stuff, gets into everything.
Getting stuff from Earth to the Moon is hugely energy intensive and expensive, so as much as possible, whatever could be obtained from the Moon, would be and as little as possible would be flown from Earth.
Basics like water, air to breath, vacuum sealed living spaces so they wouldn't need to wear pressure suits all the time, soil to grow food and machinery to build stuff (there's no point of just living on the Moon, it should be used for mining and building stuff). Some manufacturing would be easier in the low gravity situation of the Moon, and unlike the Earth, where a space elevator is enormously hard to build, a space elevator could be built from the Moon for easy and cheap launches.
I would think that the energy problem on the moon would be solvable, presumably by solar panels. Perhaps by nuclear, though that takes a lot of equipment. Solar might be easier to mass-manufacture. Then you'd need batteries, which are heavy. All that would probably be built on the Moon which would take time, but cheaper than sending material from Earth.
Some items they might be short on.
1) Nitrogen. Nitrogen is useful for fertilizer and it's useful to breathing. Breathing pure oxygen is unhealthy (or so I've heard). It may also be useful for some kinds of rocket fuel (I think I read that somewhere). There's not much nitrogen on the moon. (Mars has the same problem).
2) hydrocarbons. The Moon is dusty - really fine teeny tiny very annoying dust particles that get into everything. Oil keeps stuff lubricated and oil makes a good sealant. When (if) a colony is set up on the moon, they're going to want oil for a lubricant, and they'll want to manufacture rubber. There's no oil deposits below the surface of the moon because there was never plant or sea life that got buried and turned into oil or coal over time. There's very little carbon on the moon, no CO2, very little carbonate rock (er, I think).
I think, those are the big two, Carbon and Nitrogen.
3) Lithium. This one I'm torn on. Used for batteries, it's lower weight than Lead-acid batteries, but the low gravity of the Moon, lead-acid might not be a problem. You can't burn fuel without oxygen in the atmosphere, so that leaves batteries or nuclear powered equipment.
4) Water, and I know that there's a report that there's a billion tons of water on the Moon and that may be true, but it's underground. There's only a little water on the surface in craters that's easy to access. Water would be a valuable resource on the Moon, not to be taken lightly.
There's probably a dozen others that I'm not smart enough to know about and there's plenty of Carbon and nitrogen and water on comets and asteroids, (especially comets), so, it's not an unworkable problem. Capturing a small comet, harvesting it, while keeping it shaded from the sun, changing it's orbit - not easy. All the ins and outs of space colonization make my head hurt. It won't be easy.
biologically produced ones like coal and other petrochemicals. Marble, limestone, Diatomaceous earth, and other such minerals are candidates as well.
Manganese oxides may be a good candidate the biologically formed ones have several unique industrial applications due to having an unusual microstructure. It is also most easily mined from seafloor sediment.