Hi M.,
First, please don't give your baby ANY medicine, as others have recommended. It is NOT meant for babies at all, especially a 1 1/2 month old. This is why the FDA is taking away our medicines---b/c they've been overused and applied in situations that they are not meant to be applied!!
My baby boy was sick constantly once he went into daycare at 10 - 11 weeks old. The doctor told us the same thing that yours told you: there's not much you can do but let their own bodies heal themselves (it's good for their developing immune system too). He told us to use saline drops (a drop or two in each nostril), the use a bulb syringe to suction his little nostrils. This helped to relieve our baby of mucus (for awhile). So, we'd always do that several times a day...especially at bedtime to help him sleep just a little while longer.
About bedtime, as others have posted, the drainage and congestion in his chest (which causes the coughing) worsens when they lay down (just like it does for all of us!). So, our doctor told us to elevate the head of his crib (actually, it was a bassinet at that age) to help the drainage to not settle in our baby's chest. We were also told that using a humidifier would help, as some of the other mamas mentioned. We just never bought one, so I can't tell you much about that.
As Anu mentioned below: Keep an eye on that cough and make sure it doesn't worsen. If it does, go back to your doctor because it could be the RSV respiratory virus. My baby had that as one of his illnesses and he was coughing so hard that he was gagging and throwing up. It was sooo sad. So keep an eye out for that type of coughing!!
And, not to scare you, but I have heard that Whooping Cough (Pertussis) is making a come-back. So, that is another reason to take him to the doctor immediately if his cough worsens.
Pertusses is often passed to babies from adults, so my doctor recommended that any adults in contact with our baby, such as my husband and I, the grandparents, any caregivers, etc. get the TDaP (Tetanus, Diptheria, acellular Petrussis) vaccine to boost our prior Pertussis immunity. It can't hurt to be extra cautious at that tender age!!
Take care and stay strong!
D.