What is the full name of Ben Linus's adoptive daughter in Lost?

Ben Linus’s adoptive daughter in Lost is Alexandra Rousseau.

Alexandra, usually called Alex, is the biological daughter of Danielle Rousseau and Robert. She was born on the Island during Danielle’s French scientific expedition, roughly 16 years before Oceanic Flight 815 crashed. Shortly after her birth, Ben Linus took her from Danielle and raised her among the Others, allowing Alex to believe that her biological mother was dead.

Alex’s unusual family history creates one of Lost’s central identity conflicts. Although Ben presents himself as her father, he is not related to her by blood. Danielle is her mother, while Ben is her adoptive father. Alex eventually learns the truth and is reunited with Danielle near the end of the third season.

Played by Tania Raymonde, Alex often helps the crash survivors despite her upbringing among the Others. She assists Claire, Kate, Sawyer, and Karl, and repeatedly resists Ben’s control. Her story ends tragically in the fourth season when Martin Keamy shoots her after Ben refuses to comply with Keamy’s demands.