The problem here is in the definition of "to mine." In the North American Mining West, one frequently sees a bumper sticker that reads "If it isn't grown, it is mined" - technically everything people use that isn't organic has been mined in some capacity. By that definition, hominids started mining when they first picked up a rock and used it as a tool.

But that is likely NOT what you're asking. You're likely asking about is mining as popularly conceived (that is, going underground) rather than the way the industry perceives itself. People started going underground to mine in the prehistoric period, so we have to back into how they did that. The easy answer is that people started collecting things on the surface, and when they noticed concentrations of things they could use, they started digging pits. When the pits were exhausted and it seemed that the resource continued underground, they pursued it underground with shafts (vertical excavations) and tunnels (what the industry calls adits: horizontal excavations).

We can see something of the evolution of this process in flint mines of southern Britain. In some places there are acres of surface pit mines, but then one also sees the excavations underground into the chalk in pursuit of underground flint nodules.

A similar process occurred in Cornwall in pursuit of tin. We even see this same process as late as the gold rushes of the American West: miners began by working streams and river beds/sand bars. When these were exhausted, they began pit mining, digging down but not going underground. When this was exhausted, they began the more expensive and dangerous process of pursuing concentrated veins underground. In California, the first phase began in 1848/49. The second phase was in full swing by the early 1850s, and the third, underground process was begun in the mid 1850s, lasting for a century.