What are the best herbal home remedies for a kid with a dry cough?

Homeopathy, actually: This is not the same as an "herbal" treatment. But classical homeopathy actually has many medicines ("remedies") that stimulate healing of cough. Different kinds of coughs correspond to different homeopathic remedies, so you'd have the best chance of success by consulting a homeopath. But the remedy rumex can help a tickling, irritated dry cough that gets worse when undressing or uncovering.