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".