How do I pray for the health of a newborn?

Simply, and often. You do not need eloquence at three in the morning. Say the child's name, ask that their body grow strong and their breath come easily, and ask for a quiet night for them and for you. Add one line for yourself: that you be given the strength this asks of you. New parents pray badly and constantly, and that is exactly the right way to do it.

The old blessing over children is birkat habanim: a parent's hand on a small head, asking that this life be kept and go well — the plainest prayer there is.

From the sources “Your children will be like olive shoots around your table.” — Psalm 128