Answer: Kawasaki syndrome.
Step 1: Kawasaki disease is an acute medium vessel vasculitis of childhood. The diagnosis needs fever for 5 days or more plus at least 4 of 5 principal features.
Step 2: Those features are: bilateral nonexudative (nonpurulent) conjunctival injection; oral and pharyngeal changes (injected pharynx, red cracked lips, strawberry tongue); polymorphous rash; changes of the extremities (erythema, edema, later peeling); and cervical lymphadenopathy. The clues in the stem map directly onto these criteria.
Step 3: Measles has Koplik spots and a typical conjunctivitis with coryza and cough but not the full Kawasaki picture. Thrombocytopenia is a lab finding, not a clinical syndrome with these signs, and mumps causes parotitis. So Kawasaki syndrome fits best.
Ref: Nelson Textbook of Pediatrics.