My baby suddenly won't let anyone but her father or me pick her up. Is this normal?

This is very normal, although it can be quite disconcerting when your little one does this to a grandparent. Stranger anxiety begins at around eight-months-old and will likely subside around age eighteen months.

The timing of stranger anxiety is related to the developing personality of your baby. As her brains develops, she can discern the people and things around her. She will be "comfortable" with the familiar—her parents, blankie, bed, etc. When she is separated from the familiar, it is distressing, hence, separation anxiety. As she continues to grow and develop, your toddler will understand the permanency of mom, dad, and familiar settings and things, and the anxiety will abate.