Stone boiling is what archaeologists and anthropologist call an ancient cooking technique that involves placing stones into or next to a hearth or other heat source until the stones are hot. The heated stones are then quickly placed into a ceramic pot, lined basket or other vessel holding water or liquid or semi-liquid food. The hot stones then transfer the heat to the food. Stone boiling is a way of heating food without direct exposure to flames, which is trickier if you don't have hot pads and insulated oven mittens.
Boiling stones typically range in size between large cobbles and small boulders, and for safety's sake they should be of a type of stone that is resistant to flaking and splintering when heated. The technology involves a considerable amount of work, including finding and carting around appropriately sized stones and building a large enough fire to transfer sufficient heat to stones to make it useful.
Invention of Stone Boiling
Direct evidence for using stones to heat liquid is a little hard to come by: hearths by definition generally have rock in them, and identifying whether the stones have been used to heat liquid is difficult at best. So, we have to look at the history of hearths. The earliest evidence that scholars have suggested for the use of fire dates to ~790,000 years ago; although that is somewhat debated, and even if it was a real fire, it's possible it was used for warmth and light, not necessarily cooking.
The first real hearths date to the Middle Paleolithic (ca. 125,000 years ago. And the earliest example of hearths filled with heat-fractured round river cobbles come from the Upper Paleolithic site of Abri Pataud in the Dordogne valley of France, about 32,000 years ago. Whether those cobbles were used to cook with, is probably speculation, but definitely a possibility.
According to a recent study conducted by Nelson using a handful of ethnographic databases, the method of stone boiling is used most heavily by people who live on that part of the earth that lie in the temperate zones on earth, between 41 and 68 degrees latitude. All kinds of cooking methods are familiar to most people, but in general, tropical cultures more often use roasting or steaming instead; arctic cultures rely on direct-fire heating; and in the boreal mid-latitudes, stone boiling is most common.
Why Boil Stones?
Thoms has argued that people use stone boiling when they don't have access to easily cooked foods, such as lean meat that can be direct-cooked over a flame. He indicates support for this argument by showing that the first North American hunter-gatherers didn't use stone boiling intensively until about 4,000 years, when agriculture became dominant.
Stone boiling might be considered evidence of the invention of stews or soups. Pottery made that possible. Nelson points out that stone boiling requires a container and a stored liquid; stone boiling involves the process of heating liquids without the dangers of burning a basket or the contents of a bowl by direct exposure to fire. And, domestic grains such as maize in North America and millet elsewhere require more processing in general to be edible.
Any connection between boiling stones and the ancient story called "Stone Soup" is sheer speculation. The story involves a stranger coming to a village, building a hearth and placing a pot of water over it. He (or she) puts in stones and invites others to taste the stone soup. The stranger invites others to add an ingredient, and pretty soon, Stone Soup is a collaborative meal full of tasty things. NOt to mention a stone or two.
Sources
This glossary entry is a part of the Dictionary of Archaeology.
Nelson K. 2010. Environment, cooking strategies and containers. Journal of Anthropological Archaeology 29(2):238-247.
Thoms AV. 2009. Rocks of ages: propagation of hot-rock cookery in western North America. Journal of Archaeological Science 36(3):573-591.

