First of all let the person know that they are not alone in this struggle. Tell them there are thousands of women in the globe that suffer from the same condition after the birth of child. You will have to make them comfortable about it, so that she trusts you and share her pain and agony with you.
Being a mother is very hard, especially for a first timer. Obviously she is not used to waking by the cries of the child in the middle of the night. Sharing the burden is a good way to ease the pressure from the new mom. Divide duties on cleaning up the baby, or divide nights for looking after the new born child.
One of the most important things a mother needs is plenty of rest. Ideally the mother should sleep as much as her child and the saying “Always sleep when your baby does” speaks for itself. Turn your cell phone to answering machine, or let your husband take all your calls.
Make sure that the new born mother does as little physical work as possible. Cook her healthy meals and keep her diet high in protein and iron to recover from the effects of child birth.
Obviously mother will not allow baby sitters to hold their babies in their arms, but you can arrange for a full time maid that will look after both the mother and the child, and attend to their needs.
Talking about your experience with other women who have passed through the same stage in their lives is a good way to make the sadness go away. Let the mother some alone time so that she may cry as hard as she want to ease her pain.