Since it started yesterday he could indeed just have a cold. However nighttime coughing and coughing when running could also be a sign of asthma. It could be a variant form that only shows itself when your son has a cold. I suggest that you do see his Pediatrician for evaluation.
Milk can increase mucus so limit his milk intake if possible for a few days. Honey on a spoon can actually help quiet his cough.
When your children's cold is in the initial stages then the following at home natural remedies can help your child's body to fight back and can also aid boosting their immune systems.
1. Lots of fluids including chicken soup, tea, broth, or clear liquids can help clear out the mucus that has developed as a result of the irritants.
2. A cool mist humidifier can help with the discomfort of a scratchy throat and with stuffy noses.
3. Steam: from a shower for 10 minutes can also help relieve congestion.
4. A saline nasal wash is particularly great for children of all ages and particularly young ones, who can't blow their noses. Drops are placed in each nostril and a suction bulb pulls mucus out of the nose. Saline nasal spray or even your own concoction (a dosage syringe obtained through your local pharmacy filled with diluted saline in water) is a must for children's colds and for adults too. It helps to relieve stuffiness and thin out the mucus which ultimately promotes healing.
5. Lots of rest. The common cold can be weary on a little person's body so it's important that they get plenty of rest.
6. Vitamin C.
Hope this helps. I know how hard it is when you little one is ill,.
Of coarse if he worsens with difficulty breathing or high fever or not drinking or eating go to the ER.