What does Amy mean?
From the Old French Amee (modern Aimee), the past participle of amer 'to love', ultimately from Latin amata 'beloved'. A clear reflex of the Latin root am- 'love'.
What does Nancy mean?
Originally a medieval diminutive of Ann (from a misdivision of 'mine Ann' into 'my Nan'), carrying Ann's meaning of grace or favour from Hebrew Hannah. An older alternative traces it to Agnes ('pure'), so the grace meaning is the usual but not sole account.