Hi, OP, I've had eczema on and off since childhood, I used to have it on my arms, neck and at the back of my knees, but currently I only have a bit on my hands. I never quite grew out if it, but have learned over the years what triggers to avoid.
Laundry products - avoid anything biological, or highly perfumed. Bold and Persil both have reputations for irritating eczema prone skin. Also avoid anything that relates to restoring/protecting colours - these chemicals are very irritating to skin. Also, use less product than is recommended - you often can get away with using just half, and that way there is more chance the residue will rinse away. Or even just use washing balls (google them). I don't use fabric conditioners at all - not worth the risk and not needed, esp. if you have a tumble dryer.
I cannot tolerate ointments on my skin, as they tend to make my skin overheat. Most ointments contain petroleum jelly/AKA white soft paraffin, which is useful as a barrier to hold moisture in, but not great if your skin cannot tolerate the greasiness. Often GPs prescribe ointment as the default position (perhaps as they are cheaper?) but it may be worth asking for the cream version. Many creams contain small amounts of alcohol (perhaps as a preservative?) but not enough that the alcohol should irritate the skin. I find Aveeno cream is good for my dry face skin.
Pets - do you have any? I developed allergies to cats and dogs when I was in my late teens. Furry pets are best avoided if children are prone to allergies - it's just one more thing to cope with.
Stress - yes, defo a factor.
Good luck OP, it's very hard but maybe your 5yr old will grow out of it, as many kids do. Otherwise, remember the golden rule - the less chemicals and perfume he comes into contact with, the better. Some of the biggest brand names have the worst products, as far as containing skin irritants goes. They smell great (if you like that sort of thing), but the stronger they smell, the worse they are, IME. Also - less is best, re cleaning products generally.
Windolene - just the smell starts me sneezing, and if I get it on my hands it really irritates. Foul stuff. Air fresheners ans spay polish - just no. Too many chemicals to set me off sneezing. I wet dust, using a bowl of hot water to keep rinsing and wringing out the cloth, and buff TV screen/mirrors dry with a clean dry cloth.