How did Greece get its name?

The name 'Greek' is in fact ancient as well, as the famous philosopher Aristotelis verifies: "
...and she was not there forever, but after the cataclysm of Deucalion, which occurred in the Hellenic area, in fact, in the ancient Hellas, which was around Dodoni, and it changed many times the flow of Acheloos river. In that area live the Selloi and the ones that were once called Graecoi and are now called Hellenes..." [Aristotelis Meteorologika, I, 14]
Greek was a name used by the Romans when they first ran into the colonies (Tarentum, Sybaris, Croton etc) in southern Italy, named after the Greek tribe they knew as the Graeci who had retained the older name that Hellenes (were known by). According to the Greek tradition, the "Graeci" ("Greeks", Γραικοί, Graikoi) were renamed "Hellenes" probably with the establishment of the Great Amphictyonic League after the Trojan war.
The Greeks called themselves Hellenes after their legendary founder Helen.

As the Romans penetrated south in the Italian peninsula they ran into a tribe called the Graeci. They then ran into the Hellenic cities in southern Italy, and thinking that they were the same people in large numbers, called them Magna Graeci - Great Greeks.

The name Greek stuck. The Greeks still call themselves Hellenes.