Cities are often built near resources
The key mining/mountains example would be Potosi, a city of 100,000+ that grew up around the famed silver deposits of Cerro Rico in Bolivia, despite being in a desert at over 4000m elevation. It was by far the largest city in the Americas in the late 1500s and early 1600s.
San Francisco, in turn, saw a massive influx of people as the chief port of the gold chasing 49ers. In 1848, it had a population of under 1,000; in 1849 25,000; and in the 1850 census (conducted in 1852, for complex reasons) 34,776.
So there are ample historical precedents for such a city to spring up, literally overnight.
There are plenty of examples of minerals being mined from currently active volcanoes or recently extinct volcanoes. Some of the other answers mention diamonds. While diamonds are moved to the surface by volcanoes, they were moved millions to billions of years ago. Most of the big diamond regions of the world (Angola, CAR, South Africa, Russia) haven't seen volcanic activity in a long time. I don't think diamonds fit the bill for what you are looking for.
Lets restrict our search to things that are mined from currently active volcanoes, or at least extinct volcanoes close to currently active ones.
Sulfur is mined at the active Kawah Ijen complex in Indonesia on the island of Java.
Gold is mined at the extinct Luise caldera on the Papua New Guinean island of Lihir. This operation is somewhat complex, as geothermal vents have been cut to lower temperatures around the gold deposit so it can be mined. It is one of the largest gold deposits in the world. While the caldera itself is extinct, the region is still part of the Ring of Fire, with many active volcanoes nearby on New Britain and New Ireland.
Pure Rhenium was discovered in the active Kudriavy volcano in the Russian Kuril Islands. Rhenium isn't used in a lot of things so it isn't super expensive(thought at $85 dollars per ounce it is still 5 times more expensive than silver), but it is rarer in the Earth's crust then gold or platinum. So if it were in demand for some high tech (or magical?) application, it would be very expensive indeed.