Sorry for the long post but I feel your pain!
My DS had exactly the same thing and even now at 6yo he still suffers with horrible ongoing coughs following a cold. I was up at 3am with him just last night.
A bit of history - when he was 2yo he had a chest infection which he had 3 courses of antibiotics for, but even after that the cough continued. It started in the November and the cough went on until about March, when the weather started to improve. But since then every time he gets even a minor cold it always results in a cough - sometimes very dry and irritating and sometimes very loose and chesty sounding. This went on and on and we exhausted every possible avenue to try to figure out why he suffered so badly each time. He had chest x-rays, inhalers of all colours for asthma, asthma medication including Singulair, reflux medicine, a Cystic Fibrosis test (negative thankfully) and the list continues. We paid to see a private chest and lung pediatrician at the Royal Brompton hospital but even then he was a bit of a mystery to the medical profession. So, in the end we had to put it down to 'just one of those things' and accept that he is just a kid who seems to suffer with coughs.
It has got a bit better as he has got older, the first couple of years he had five or six bouts of uncontrollable coughing, day and night and it lasted for weeks at a time. The year when he started full time school was horrible as he picked up cold after cold which resulted in him having a permanent cough for months at a time. He missed a lot of school that year as, while he wasn't actually ill with it, he would be awake half the night coughing and be too exhausted to go in, or I would get a call from the school telling me that he had fallen asleep on the carpet again. I am hoping that, as his immune system gets stronger he will pick up less colds and his body will be able to deal with it better.
There are a couple of things I can recommend though - as long as you have seen the doctor and ruled out anything serious, obviously - a blue inhaler really seems to help when he has a dry cough as it helps to open up the airways and makes it more loose and productive. You can ask the doctor for a spacer to administer help the dose. The second thing is that I give him 5mls of Allitech Liquid each day in some juice. It is a pure form of liquid garlic and it helps strengthen the immune system and warn off germs. You can order it via a website specialising in alternative and herbal medicine called Dulwich Health. It's not cheap at £49 a bottle but it does last quite a while. Neither actually cure the cough but they have both helped a lot.
Oh, and honey (particularly Manuka honey) is ten times better than any cough syrup.
I know how exhausting and worrying it is to see your LO coughing his heart out night after night. It's just horrible and so frustrating to not know why or how to help them. Big hugs to you all and I hope if nothing else that my post has made you feel that you are not alone.