What can I do about my 10yo's eczema which started as rash around corner of mouth now spread to cheek and back of neck? Prevention and treatment?

Determine cause. One of the most important elements in treating eczematous rashes is to determine the cause if possible. See an allergist or dermatologist for evaluation. If needed, allergy testing should be performed.
Secondary infection? Cracking of skin at the angles of the mouth, tho not eczema, is very difficult to treat because it 'recracks' each time the mouth opens. Vicious cycle! Keep it elastic & supple with absolutely continuous and constant lip balm, vaseline, etc. Until it heals. If he has eczema & it has 'exploded' recently, there may be a secondary bacterial infection, and a trip to the doctor may be worthwhile.