Is Annie's mother. On the very last page in the book Carrie that they are reading in the book-club, there is an excerpt from a letter sent by a woman named Amelia to her sister. She explains that her daughter, Annie, seems to have telekinetic powers.
Named for Amelia Earhart, famed female pilot that went missing somewhere over the Pacific ocean.
She is actually Amelia Earhart, who would have been the survivor of the very first plane crash on the island. Although born over 100 years ago, the island's healing energy may have kept her young. Or, time moves differently on the island.
This theory is further fueled by various hints from chapter 2 of Find 815.
Or she is named Amelia because Earhart crashed on the island, she could even be Earhart's daughter.
She is pretending to be Amelia Earhart.
Amelia is Amy
Like Ellie was a short name for "Eloise", "Amelia" is Amy, the wife of Horace Goodspeed.
According to the Lost Encyclopedia's Ethan article Amy did in fact die in the Purge.
After seeing something DHARMA is doing (possibly attacking the time traveling survivors after Phil gets captured) Amy get angry at Horace, takes Ethan, and joins the Others. After that she decides to go by Amelia instead of Amy.
Very possible. The tone of the dialogue she has with Ethan could indicate she is his mother, Amy.