Surprisingly enough, I've never really been much into the Amelia Earhart "mystery," mostly because I think it's clear what happened to her. Earhart's plane ran out of fuel, crashed into the vast Pacific Ocean on July 2, 1937, and was lost. Given the expanse of the Pacific Ocean, and the lack of land in the area where Earhart was flying, it is entirely possible that no debris from her crash would ever be found.
Nevertheless, theories have abounded over the many decades about her disappearance, from Japanese capture and torture to assuming another identity and living in Asia. Since her body was never found, there's always the possibility she survived the initial crash. Also, there's always the question of what happened for sure.
While most stories can be dismissed as mere fiction, there are a couple of realistic possibilities of what may have happened to Earhart. Assuming that she could have landed the plane in water or on land safely (or survived the impact) there is an island called Nikumaroro in the area where Earhart was last known to be. Supposedly, in the early 1940's British surveyors found skeletal remains on this island, though these tantalizing remains have been since lost to obscurity.
Lately, a research group called International Group for Historic Aircraft Recovery (TIGHAR) has been re-examining the Amelia Earhart mystery, and Nikumaroro Island. Its executive director Ric Gillespie postulates that Earhart was a castaway on Nikumaroro Island and even sent radio distress calls days after her disappearance.
In 2007 TIGHAR found artifacts which do raise interesting questions. The items, a mirror compact and a zipper, have been authenticated to be American-made objects circa 1930's. TIGHAR's plan is to return to Nikumaroro and hopefully excavate more artifacts. If successful, researchers hope to extract DNA from the items, and compare them to a reference sample of Amelia Earhart's DNA. It's a longshot, but I suppose it's possible.