The main minerals in tap water are calcium and magnesium, which determine the hardness of water. The main minerals in mineral waters are calcium, magnesium, sodium, chloride, carbonate and sulfate. Tap water from different regions and various brands of mineral water can contain very different amounts of minerals, so it is impossible to make a general recommendation to drink more mineral water.
Mineral water high in calcium or magnesium can be a good source of calcium and magnesium for people who do not get enough calcium from foods. Foods are much better source of calcium and magnesium than water, though.
Researchers from the World Health Organization (WHO) claim that water hardness (calcium and magnesium content) has no effects on health (positive or negative).
Foods high in calcium, beneficial and harmful effects of calcium and references to claims I made:
Calcium concentration in bottled and mineral waters (US, Europe):