A rash around the mouth suggests a food cause. Def start by cutting out eggs, but be aware it could take awhile to see improvement. Also, he could grow out of it, so it doesn't mean no eggs forever either.

The itchy head sounds just horrible for him. It's easy to be distracted when awake, so he might not itch when there's other stuff to hold his attention. DH still wakes me occasionally at night if my eczema has flared up because I'm itching in my sleep!

Sounds like you definitely need steroids/cortisone to get on top of it, then it's a matter of trial and error in finding a gentle moisturiser to slather on several times a day. I'd be seeing your GP again, maybe a specialist if you can't get on top of it. Lots of people are singing the praises of coconut oil at the moment - might be worth a try.