This looks very much like an allergy. It is probable that he has developed an allergy, ie, contact dermatitis, to something, you should check out whether you have started any new soap, ointment or detergent within this period. It could be from some new article of clothing or jewellery. They might also be due to some new medications that he is using.
It could also be due to some food that he took very recently. Watch out for foods like shell fish, sea fish, meats, cheese, etc. Also any new types of processed foods. If you do find anything which could be the cause, then try to avoid it.
Next, he can take an anti-allergic medicine, an antihistamine. But for any use of antihistaminic before 2 years of age, you should first discuss with his doctor, give him only after discussing with his doctor. He can take Benadryl elixir for children, an OTC anti-allergic medicine. The dose is about 0.5 mg per pound of bodyweight (or 1.25 mg per kilogram). For example, a 25 lb child would get one teaspoon of liquid or one chewable tablet every four to six hours.. Also, apply a soothing lotion like calamine lotion, also available OTC.
Another possibility is bug bites like bed bugs, mosquitoes, spiders, etc.. In any case the treatment would be the same.
The other likely cause, if he been exposed to much heat, is Prickly heat or Miliaria. This is essentially a reaction that occurs due to heat. The treatment for this is the same.
There are various other possibilities like scabies, eczema and other dermatological conditions which might cause similar spots, but in his case they are most likely to be allergy. So you can try out these two steps.
However, if new spots keep on appearing after you have tried this out for about 10 days, then you should approach your doctor and get it examined.