Step 1: Anchor on the single most discriminating fact - the family history of asthma. Asthma, hay fever and eczema form the atopic triad, so a child whose family has asthma is far more likely to have the eczema of atopy.
Step 2: Atopic dermatitis fits this profile: an intensely itchy, red, weepy rash in a child, favouring the cheeks and limb flexures, running in atopic families. The asthma history is essentially a built-in pointer to this diagnosis.
Step 3: Weigh the others against the history. Allergic contact dermatitis depends on touching a sensitiser and stays where the trigger touched, unrelated to inherited asthma. Seborrheic dermatitis is greasy and scaly over sebum-rich zones, and erysipelas is an acute streptococcal skin infection with a raised, sharply bordered, painful red plaque and fever. None links to atopy the way the asthma history does. \[\boxed{\text{Atopic dermatitis}}\]
