Here's the story:
Claire's birth mother, Meredith Gordon, was a woman with the ability to heal herself and others from any injury. This power was inherited by Claire. However, Meredith was unstable and struggled to control her abilities.
To protect her daughter, Meredith left Claire with a couple in a fire station after Claire was born. She gave Claire a necklace with a locket containing a picture of Meredith and a note about her powers.
Meredith's fate is never explicitly revealed in the series. However, in the Season 2 episode "The Second Coming," Claire discovers her mother's past and learns about her powers. Meredith is later mentioned in the Season 4 episode "The Fifth Stage," where she is said to be living in a "secret compound" and is trying to find Claire.
Ultimately, Meredith's fate remains a mystery in the show. However, the implication is that she is still alive and seeking her daughter.