Yes, the original Greek word “tragōidia” literally meant “goat song.” There are several theories as to why this is the case.
A theory popularized by Aristotle is that “song of the goats” refers to the chorus of satyrs in dithyrambs, hymns sung in honor of Dionysus, which were the precursors of Greek tragedy.
A theory inspired by Horace’s Ars Poetica suggests that the goat was a prize won for a skilled tragic performance.
Others have suggested that the original root “tragos” was referring to a type of grain used in making beer (assuming that Dionysus was originally a god of beer, not wine) or that the root word was “tragizein,” meaning “adolescent voice-change.”
Sources: Wikipedia, Wiktionary, & Oxford English Dictionary