I make my own home remedy cough medicine by mixing 2 teaspoons of honey with herbal tea and lemon.
You can also just mix the honey and lemon with warm water if you don't have any herbal tea.
The honey in the herbal tea or water along with the lemon will make a good cough suppressant and helps to get rid of a nagging cough.
Taking steamy hot showers or sitting in the bathroom with the shower steaming is also good for getting rid of a cough and using a humidifier or vaporizer in your home to add moisture to the air can also help get rid of a cough.
Sipping on hot drinks throughout the day can also rid you of the cough.