For a cough related to the common cold...
First, note that these are to treat common colds, and it's important to rule out more serious illnesses.
I steal some of my favorite treatments from this article from Dr. Sears. I've posted the link so you can read for yourself, but I've revised or annotated the article to the things that I personally use for my daughter.
1. Steam cleaning. Give your child steam, steam, and more steam. For infants and young children, turn the bathroom into a steam room with the door closed and the shower on full hot. Sit in there for 10 or 15 minutes. Do this steam cleaning every morning and before bed, as well as during the day if possible.
2. Clap the chest and back. While you sit in the bathroom steaming, clap on your child’s chest and back (where the lungs are) firmly (harder than burping) with an open hand. This helps shake the mucus loose so your child can cough it up better.
3. Sleep upright. If possible, allow your child to sleep in a slightly upright position. This allows for easier breathing during sleep.
4. Nose hose. For older children, it is crucial to have them blow their nose several times during a steam cleaning, as well as frequently throughout the day. Getting out all the junk will help prevent this from turning into a bacterial infection. An alternative to steaming is to use nasal decongestant spray to loosen up the nasal congestion before blowing it out. For infants too young to blow their nose, you can suction them out using a blue rubber bulb syringe. Or try the Nosefrida!
5. Hot steam vaporizer. Use a hot steam vaporizer in the bedroom at night (not a cool mist humidifier). This warm, humid environment can help keep noses and chests clear at night. Be sure to air the room out well during the day because mold can start to grow in the room due to the warmth and humidity.Dr. Sears suggests: To prevent burns keep your hot mist vaporizer out of reach of infants.
I actually prefer a cool mist humidifier for safety reasons--the Mayo Clinic has a good explanation: "...by the time the water vapor reaches your child's lower airways, it's the same temperature regardless of whether it started out warm or cool."
We invested and bought this humidifier--it has multiple functions, tracks the moisture in the air, and is super-quiet:
6. Drink drink drink!