A rainbow baby is a baby born after a miscarriage or any kind of loss...
A "Rainbow Baby" is the understanding that the beauty of a rainbow does not negate the ravages of the storm. When a rainbow appears, it doesn't mean the storm never happened or that the family is not still dealing with its aftermath. What it means is that something beautiful and full of light has appeared in the midst of the darkness and clouds. Storm clouds may still hover but the rainbow provides a counterbalance of color, energy and hope.
The cool thing is that we actually have two "Rainbow Babies". We had a miscarriage before Josephine and two before Beatrice. I've always struggled with a way to characterize births after losses, and this couldn't be a more perfect way to look at it.