Every day, Judy Carr enters her living room and considers the bags of small plastic toys accumulating on her coffee table, abandoned one by one by her 5-year-old son after the latest birthday party.

NEW YORK — Anya Roles, 34, a Manhattan mother of three girls under the age of 5, has seen a range of gifts at children’s birthday parties over the years: personalized lunchboxes and pricey plate sets, ...