What was the name of the French woman who had survived on the island before the crash?

The French woman who had survived on the island before the crash was Danielle Rousseau. She arrived there sixteen years earlier as part of a French scientific expedition and became known to the Oceanic 815 survivors simply as “the French woman.”

Rousseau’s research vessel was drawn toward a repeating radio transmission containing the numbers 4, 8, 15, 16, 23, and 42. The ship ran aground near the island, where Rousseau and her pregnant crew became stranded. She later gave birth to her daughter, Alex, but Alex was taken by the Others.

Rousseau claimed that her crewmates contracted a mysterious sickness, leading her to kill them because she feared the illness could spread. She then lived alone for sixteen years, building traps and monitoring the island. Her first major encounter with the survivors came when she captured Sayid and warned him about the Others.

Her long isolation and search for Alex drive much of her story. She is eventually reunited with her daughter, although the reunion is brief: Rousseau is killed by mercenaries during the fourth season.