Ethiopia's name is Greek, and for most of antiquity it did not mean Ethiopia at all. It meant almost everything south of Egypt.
The root is the Greek Aithiops. The Greeks heard it as a compound of aithein, "to burn," and ops, "face." The Perseus Digital Library, quoted by Wikipedia, glosses the noun as "burnt-face" and the adjective as "red-brown."
Homer is the first writer to mention the Aithiopes. He puts them at the far eastern and western edges of the world, where the sun rises and sets. They appear twice in the Iliad and three times in the Odyssey.
The word did not yet point to one country. Around 440 BC Herodotus, who traveled up the Nile as far as Elephantine, used it for the peoples south of Egypt. He most often meant Nubia, and he describes a capital at Meroë. When the Hebrew Bible was translated into Greek around 200 BC, the land of Kush became Aethiopia too.
The name moved to its present home through conquest. In 330 AD King Ezana of Aksum conquered Meroë and took over the Nubian title "king of Ethiopia." From then on Byzantine Greeks called the Aksumites Ethiopians. Later rulers, including Amda Seyon I and Zara Yaqob, used "Kingdom of Ethiopia" themselves.
Europe kept using the word loosely. English borrowed Ethiop in the late 14th century from Latin Aethiops, which Etymonline glosses as a general word for a Black African.
English also had a second name for the kingdom, Abyssinia, recorded from the 1630s. Etymonline traces it to Arabic Habasah, the name of the region.
How far to trust "burnt face" is a fair question. Etymonline describes the link with aithein and ops as a popular etymology rather than a proven one. It compares an adjective aithops, "fiery-looking," later "sunburned." Whatever its deeper source, burnt face remains the traditional reading.