Euphonious derives once again from those Ancient Greeks. While the saying warns about accepting gifts from Greeks it is fortunate we have accepted and adapted many useful words from them. In this case euphonious derives from the Ancient Greek word εὔφωνος euphōnos, meaning "well-sounding" or "sweet-voiced" (εὖ eu means "well" or "good" and φωνή phōnē means "sound", hence "of good sound").

And that leads us to that brass musical instrument, the Euphonium. I'll leave you to decide how "sweet voiced" it is.

The word euphonious was first used in English in 1774. It comes from the Late Latin word euphōnia, which means "sweetness of voice". 

Euphonious is an adjective that means something is pleasant to the ear or agreeable in sound. For example, you might describe a voice as euphonious. 

Euphony was first used in English in themid-15c., from French euphonie, from Late Latin euphonia, from Greek euphonia "sweetness of voice," related to euphonos "well-sounding," from eu- "good" (see eu-) + phone "sound, voice," from PIE root *bha- (2) "to speak, tell, say."  It is related to: Euphonic (1782); euphonical (1660s); euphonious (1774).

Just the facts:

Euphonious Definition & Meaning - Merriam-Webster