A.L.
could be Dustmites... my son is allergic to them, we have encased his mattress/box spring with plastic lining that you can buy. It's helped somewhat, but not completely. Additionally, when my son was little (he is now nine) he too had childhood Asthma, but it's since gone away but NOT his allergies, which when little, would make the Asthma worse..
Consider this.. you might be bringing in pollen from the outside.. pollen can get in your hair and clothes and IF it does, can be transferred to your pillows and other material things within in the house .. also, don't discount pets bringing in pollen into the house by way of their fur.. I too cough a lot , especially lately with hayfever and when I lie down, it's much worse... same goes for my son.. I take Zyrtec as does my son, which has helped somewhat.
I would change the pillowcases every night (to see IF this helps matters) additionally and IF possible , see about propping up yours and your child's pillow slightly. I don't mean to where you are sitting right up of course, but a slight slant, this way.... your sinuses won't drain into your throat as much.. sometimes, the coughing is caused by post-nasal drip.. again, the Zyrtec has helped ease some of the coughing .. but do check with your doctor first..
best of luck