The name India literally derives from the Sindhu river (aka the Indus river), which became Hindu in Persian (S->H is a common sound correspondence between Sanskrit and Persian), and dropped the initial H when getting into Greek. It was also used to refer to some areas beyond India -- (i.e, modern Indonesia or the "East Indies", but the origin of it was from civilizational India.).

I've never heard of India being used to refer to Ethiopia. Ethiopia itself is a phrase of Greek origin, and dates back at least to Herodotus.