The question is rather straight-forward. I consider a "traditional" tombstone as some sort of monument having an inscribed name and date of birth/death or age.
Ancient Egypt is thought to be the origination of the funeral tombstone, however the original name was Stele/Stela, and in plural stelae, comes from the latin, to stand. Stelae have been documented back to the first dynasty - 2890 B.C., they were used later by many number of other cultures. The highest recorded use is by Attica in Greece, example would be the Greece marble funerary stelae. The Roman uses of stele came from their expansion into Greece, in which they adopted some of the culture.