Step 1: Chronic, relapsing, intensely itchy, dry and inflamed skin in a young child is the hallmark of atopic eczema.
Step 2: The condition typically starts very early in life and waxes and wanes over years, with flexural involvement and lichenification from repeated scratching.
Step 3: The decisive hint is the atopic background - an asthmatic mother - because eczema runs with the atopic triad of eczema, asthma and allergic rhinitis.
Step 4: Lupus would be photosensitive and systemic, simple erythema is just redness without chronicity, and toxic epidermal necrolysis is an emergency with sheet-like detachment, so all are ruled out.
\[\boxed{\text{Atopic dermatitis}}\]