Step 1: The key clinical clue is a child with a rash plus a family history of asthma. Asthma is part of the atopic triad, which also includes allergic rhinitis and atopic dermatitis. A positive atopic family history strongly favours atopic dermatitis.
Step 2: Atopic dermatitis in children typically presents as an itchy, erythematous, sometimes oozing eczematous rash, classically over the face and the flexures, and is closely linked with a personal or family history of asthma and allergic rhinitis.
Step 3: The distractors are less consistent. Allergic contact dermatitis needs a specific external allergen contact and is confined to the contact area, with no link to an asthma family history. Seborrheic dermatitis gives greasy yellow scales over sebaceous areas. Erysipelas is a sharply demarcated, hot, tender bacterial cellulitis of the dermis with fever, not an atopic eczema. Given the asthma family history, atopic dermatitis (option b) is the best answer.