She may have re-taken her maiden name or an alias following her separation from Pierre and moving to the States with Miles. This could be why Miles' last name is Straume and not Chang.
In Superman lore, Lara was the wife of Jor-El of Krypton, and mother of Kal-El, who grew up to be Superman on Earth. Lara and Jor-El sent Kal-El to Earth to allow him to live when it was obvious to them that if he stayed on Krypton he would die with everyone else. On the Island, after Pierre Chang was told that everyone from DHARMA who was on the Island on December 19, 1992 would be killed in the purge, he sent Lara and baby Miles off to live.
Lara is sick and dying possibly from cancer. This was brought on by the same experiments on the island that gave her son special abilities. Her proximity to her husband (and therefore his experiments) made her sick.