This paper explores the co-design of an Interactive Digital Narrative (IDN) aimed at supporting the development of executive function (EF) in young children through culturally grounded interactive storytelling. Adopting a user-centered design approach, the research followed a five-phase iterative process involving nine participants from various disciplines, including social sciences, education, and Artificial Intelligence (AI), as well as parents. Three co-design workshops were conducted: (1) to integrate Swedish culture and values into the IDN, (2) to embed EF-stimulating interactions based on parent-child play, and (3) to conceptualise an AI-enhanced support tool for parent-child storytelling and cognitive development. Thematic analysis was employed to extract design insights from the audio recording of the workshops. These insights informed the development of a high-fidelity prototype featuring culturally resonant story elements that can support the development of EF in young children. Additionally, by positioning parents as companions rather than instructors and by embedding EF-supportive design into a joyful storytelling context, this work provides a framework for designing IDNs that are both pedagogically effective and playful for young children. This paper presents initial results of co-design of this IDN.