The following quotation indicates the origin of the name Aemilius:
"Numa gave to one of his four sons the name of Mamercus,
which was the name of one of the sons of
Pythagoras;
whence, as they say, sprang that ancient patrician family
of the Aemillii, for that the king gave him in sport the
surname of Aemilius, for his engaging and graceful manner
in speaking". --Plutarch's Life of Numa Pompilius
(
John Dryden's translation)
"
Mel" is a root connected with both "music" and "honey".
Aemilius (or
Aemilia) means "s/he from whom music or honey come forth", i.e., "s/he of sweet or musical speech".