Amelia Earhart was one of the first female pilots in Earth history. She had been on expeditions all over the world, and recalled seeing people doing all kinds of strange things to their bodies. In the mid-20th century she became famous for being the first woman to fly across the Atlantic Ocean. In 1937 she attempted to fly around the world, and on July 2nd, she and her navigator, Fred Noonan, took off from New Guinea and headed east, around the equator.
However, while over the Pacific, the plane ran low on gas. They began looking for an atoll to set down on, and tried to send out an SOS. Suddenly, a they saw a huge light behind them. The plane stopped dead, and then started moving backwards towards the light. That was the last Earhart remembered of the event. They were in fact being abducted by an alien species, the Briori. To the outside world, it appeared that the plane just vanished somewhere in the South Seas.
Unbeknownst at the time was that the mission was financed by her government and was part of an intelligence operation to gather information about the Japanese. Rumors about this emerged after the Pearl Harbor attack in 1941, and it later became part of established history.
Lots of people spent years or decades trying to solve the mystery. The most commonly held theory was that the plane had just crashed. But no wreckage was found despite numerous searches, so more speculative scenarios started emerging. Some thought she'd been shot down and captured by the Japanese Navy. Others thought that she and Noonan had flown off together on some sort of romantic adventure. The most ridiculed theory was that she had been captured by aliens. By the 22nd century, the mystery was compared to the the the disappearance of Judge Crater and the fate of the Terra Nova colony. (VOY: "The 37's"; ENT: "Terra Nova")
In 2371, the crew of the USS Voyager discovered that Earhart and Noonan had, in fact, been abducted by aliens known as the Briori, along with over three hundred other people in 1937, to be utilized as slave labor on a planet in the Delta Quadrant. The Voyager crew discovered Earhart and Noonan, along with six other humans, in a state of suspended animation, the only abducted Humans who had been left in stasis. By this time, the Humans had long since overthrown their Briori masters, and their descendants had established a thriving community on the planet. Upon being revived by the Voyager crew, Earhart and the others, who had come to be known as "the 37's" by the Human colonists, were given the option of joining the crew on their long journey back to the Alpha Quadrant. Earhart and the others declined, however, and elected to remain on the planet with the other Human colonists. (VOY: "The 37's")