First, a definition of "mineral" is required. A mineral can be defined as the result of a geologic process that is normally crystalline. Iron ore is not a mineral, but is a rock that is composed of iron compounds that are minerals. Gemstones are minerals.
Then, there is the issue of "naturally occurring" or "man made. There are, of course man made minerals that are rare because of the complexities, expense, instability, or limited ability to create them in significant quantities.
Meterorites, which are completely alien to the Earth, are rare minerals. Among the rarest are carbonaceous chondrites which, unlike the more common iron based meteorites, contain carbon and even some of the building blocks of organic matter. Their rarity comes from being the oldest and the most susceptible to extreme weathering over the billions of years.
Next, a definition of "rare" is required. Rarity can come from either the difficulty of getting to or making the mineral; or rarity can come from the rarity of the mineral in relation to the rest of the components of the Earth's crust or even the entire mass of the Earth.
Finally, there are minerals that have never been discovered, minerals that are so unstable that they do not last for long, or minerals that have "half lives", even though they are the oldest substances on Earth, and are greatly changed in nature since their origins.
"Rarer than that are probably reduced iron crystals from the reduction of ultramafic lava with carbonate minerals, quite simply because they have such short lifespan before becoming rust." ("Orleander", Sci Forums, 2007)
The ten rarest gems/minerals are believed to be Jeremejevite, Painite, Black Opal, Red Beryl Emerald, Musgravite, Grandidierite, Serindebite, Red Diamonds, Serendibite and Jadeite.
Painite, for example, is named after Arthur C. Pain, a British geologist who discovered it in Myanmar in the 1950s. Until the current number of 25 crystals, only three small examples were found.
Meteorites are rare because of two things: their alien nature, since they are not of this Earth; and the age and decomposition due to the iron content. The rarest of meteorites, however are carbonaceous chondrites, which are said to have the building blocks of life in them.
Of the man made rare minerals, there is Californium, Chernobylite, Technetium, Californium is a man made substance that was originally synthesized at the University of California at Berkeley in 1950. It has the atomic number of 98 and the status of a metallic actinide. Technetium is mostly man made, but there are minuscule natural occurrences.
Jadarite is the stuff of both nature and of fiction. The chemical properties of the fictional Kryptonite that was such a hazard to Superman was very close to the structure of Jadarite, which achieved great fame as a result of the fictional connection. Jadarite is a rare natural mineral containing boron and lithium. If jadarite could be found in sufficient quantities, it would be helpful for many applications.
Blue John is an example of rarity because of specific type and location. Flourite is a halide mineral with calcium flouride. Blue John is a purple/blue species of flourite that is only found in Blue John Cave in Castleton, Derbyshire, England.
Iridium is one of the rarest substances in the Earth's crust. Only a few tons of iridium are mined annually. The substance is, however, useful for many applications.
The existence of even more rare minerals is an ongoing mystery, since many of them have not been discovered yet! As always, geology is fascinating, glamorous, and even truly expensive!