She named him Aaron in the island reality. She has yet to name him in the LAX reality.
Technically thats true but. From lostpedia : Episode 3 Season 6 "What Kate Does"
Kate drives her to the hospital, where Doctor Ethan Goodspeed asks Claire if she is ready to have her baby, which confuses her. It appears the baby is ready to be delivered, even though it is still slightly premature. Ethan then tells her she can have the baby now or he can give her drugs which are safe and may stop the labor. Claire decides she isn't ready and she will take the drugs. When a monitoring device loses the unborn baby's heartbeat,
Claire panics momentarily, crying out, "Is Aaron okay?" Ethan uses an ultrasound to show that the baby had simply changed position and appears to be healthy.
Claire gives Kate her credit card.
Later, Detective Rasmussen and Officer Westly enter the hospital room, searching for Kate. Claire covers for her, claiming she was just a cab driver who helped her into the hospital. After the police officers depart, Kate emerges from an adjoining room and thanks Claire for her help. When Claire asks Kate what she had done, Kate asks "Would you believe me if I said I was innocent?" Claire says she would and, in gratitude, gives Kate her credit card.
Claire says she used the name "Aaron" because she "knew it or something," and Kate tells Claire she should keep him. The two wish each other luck as Kate departs.