There are several processes that can cause a mineral to precipitate from solution. You mention one which is adding elements to the solution This could occur in a number of ways. For example, aragonite is a form of calcium carbonate with a higher solubility at low temperatures than the mineral calcite that makes up limestone. So it can dissolve and cause calcite to precipitate. Ions adsorbed on the surface of minerals like clay minerals can exchange with ions in solution changing the concentrations and causing precipitation. Two different compositions of water can mix to cause minerals to precipitate.
Changing temperature can also make minerals precipitate.
Changing the pH or oxidation state affects the solubility of various minerals. Increasing the pH will cause precipitation of calcite from a saturated solution. One way this can occur is to degas carbon dioxide from the water. That's what causes the formation of stalactites in caves.