Headstones are still occasionally made of marble, but it is quite soft, as rock goes! When they used it almost exclusively in years past, it was because it was soft enough to carve by hand.
Many of the older headstones made of marble look like the carving is disappearing, and it is! When the polish begins to weather, so does the carving. It is a denser form of limestone and is a 3 on the hardness scale.
Bronze is also used for headstones, but loses it's finish after a period of time and forms a green patina. It is a chemical reaction of the copper in the alloy to the atmosphere.
Granite is the rock of choice for headstones. Granite is an igneous rock that forms deep within the mantle of the earth, due to heat and pressure, and is usually made up of a number of different minerals such a quartz, mica, and feldspar. Granite is recognized by experts as one of the most versatile and durable natural materials on the Earth. It is dense and is a 7 on the hardness scale. In most cases, even the best quarries provide only 10 to 30 percent of memorial quality granite. The rest of the usable material is fabricated in to other less demanding purposes such as building cladding, counter tops and tiles and pavers. Make sure you are getting monument-grade granite.