In order to determine the best herbs for you it's important to know your dosha.
Do you have a Vata, Pitta, or Kapha constitution?
Is your cough a dry (unproductive) or wet (productive) cough?
Warming/drying herbs dispel Kapha, but increase Pitta conditions.
Cooling/moistening herbs can increase Vata unless a warming herb is included in the formula.
Moistening herbs can increase Kapha unless some drying herb is included.
That being said, and once you determine your constitution the following may help.
Drying expectorants: cardamom, cinnamon, cloves, ginger, horehound, hyssop, mustard seeds, and sage.
Moistening demulcent herbs: chickweed, comfrey root, flaxseed, licorice, marshmallow, slippery elm, solomons' seal.
Cough relieving herbs: bayberyy, coltsfoot, eucalyptus, grindelia, horehound, mullein, osha, thyme, wild cherry.
Some of the above herbs can be toxic, i.e. wild cherry. There can sometimes be drug/herb interactions with licorice and blood pressure meds. As always it is best to check with a dependable health care practitioner.
Be Well,
Theresa Nartana