Blueness of the skin (cyanosis) can be a sign of low oxygen in the blood, which in turn can be a sign of problems with breathing, the heart or the blood.
On the other hand, it is, in fact, very common for normal children under a year or two of age to have periods during which their hands, feet and the area around their mouth, eyes and the bridge of their nose turns dusky blue. While older children become blue in the hands and lips when they are cold, babies may go blue in these areas when they are quite warm and for no apparent reason. This is thought to be caused by immaturity of the part of their nervous system that controls temperature. The medical term is "vasomotor instability", which is a rather frightening name for a normal and harmless finding. This sort of blueness is also common during the rising phase of a fever. During these episodes the oxygen level in the blood is normal, and there should be no signs of difficulty breathing, dizziness or fainting. In addition, the child should not be blue over the entire body - the color change is generally limited to the extremities and around the mid-face.
If there is any question whether an infant with cyanosis is having any sort of medical problem, the usual steps are to have a careful medical examination with measurement of the oxygen level. If the exam and oxygen level are normal, and there is no evidence of any breathing problem, there is rarely cause for concern.