Amelia derives from the Old Germanic element amal, meaning "work" or "industriousness" — the root that gave Old English the word for "to strive." It is unrelated to Emilia, despite the similar sound: Emilia comes from the Latin Aemilia, while Amelia is purely Germanic.
Amelia was popularised in Britain by the Princess Amelia Sophia (1711–1786), daughter of George II. The name was used steadily through the Victorian era and then declined for most of the twentieth century.
Amelia Earhart, the pioneer American aviator who disappeared over the Pacific in 1937, gave the name a quality of bravery and modernity that has never quite left it.
In Pythagorean numerology the letters of Amelia reduce to 5, The Seeker. This is a traditional interpretive system, not a factual claim about the name.