blue lips can mean a heart problem or circulatory issue. It often denotes lack of oxygen thru the body and needs to be seen by a doctor as you have already been told. Sometimes when a child is struggling with feeding and breathing can become an issue if they have not gotten the eat/breathe pattern firmly down. See the doctor if you have not already!
Definitely get her to a doctor! If it comes and goes, it could be Raynaud's syndrome.
This is a problem that needs to be addressed in the emergency room. It's a 911 call. Blue lips indicate a lack of oxygen being transported. The most common cause is simple anemia. In any event a pediatric specialist should be seen immediately. Usually the cause can be identified at the ER and appropriate treatment provided, but blue lips are nothing to fool around with.
Might be something really wrong, go to the emergency room.