I find the Indian mint a very effective home remedy for cough especially when there is phlegm.
Take some mint leaves, wash, add some water and then grind the leaves in a blender or a hand blender. Once blended, strain it and you will have a dark green mint water. You can add some honey to make it more palatable. Add enough water to drink it several times during the day.
It makes a very soothing drink and can also be given to kids. How thick the concoction depends on how many mint leaves you use. Initially you can use 4-5 medium sized leaves while you are getting used to the mint taste.
The plant is quite a hardy plant and easy to propagate. Just cut off the stem and replant it in the soil. Choose the more woody stems rather than young shoots.