Can a child get eczema on penis?

Not likely: Eczema is not common on a penis. More likely it would be seborrheic dermatitis, yeast infection or tinea-fungal infection. To cover the most likely possibilities you can buy Hydrocortisone cream 1% and Clotrimazole cream, mix a small amount of both on the fingers and apply to the rash three times a day for a week or so. That would treat all three.