Just-in-time adaptive interventions (JITAIs) in mobile health are an intervention design that provides behavior change support based on an individual's changing and dynamic contextual state. However, few studies have documented how end users of JITAI technologies are involved in their development, particularly from historically marginalized families and children. Less is known for public health researchers and designers of the tensions that occur as families negotiate their needs. We aimed to broaden our understanding of how historically marginalized families are included in co-design from a public health perspective. We sought to address research questions surrounding JITAIs; co-design; and working with historically marginalized families, including Black, Indigenous, and people of color (BIPOC) children and adults, regarding improving sun protection behaviors. We sought to better understand value tensions in parents' and children's needs regarding mobile health technologies and how design decisions are made. We examined 2 sets of co-design data (local and web-based) pertaining to a larger study on mobile SunSmart JITAI technologies with families in Los Angeles, California, United States, who were predominantly of Latinx and multiracial backgrounds. In these co-design sessions, we conducted stakeholder analysis through perceptions of harms and benefits and an assessment of stakeholder views and values. We open coded the data and compared the developed themes using a value-sensitive design framework by examining value tensions to help organize our qualitative data. Our study is formatted through a narrative case study that captures the essential meanings and qualities that are difficult to present, such as quotes in isolation. We presented 3 major themes from our co-design data: different experiences with the sun and protection, misconceptions about the sun and sun protection, and technological design and expectations. We also provided value flow (opportunities for design), value dam (challenges to design), or value flow or dam (a hybrid problem) subthemes. For each subtheme, we provided a design decision and a response we ended up making based on what was presented and the kinds of value tensions we observed. We provide empirical data to show what it is like to work with multiple BIPOC stakeholders in the roles of families and children. We demonstrate the use of the value tension framework to explain the different needs of multiple stakeholders and technology development. Specifically, we demonstrate that the value tension framework helps sort our participants' co-design responses into clear and easy-to-understand design guidelines. Using the value tension framework, we were able to sort the tensions between children and adults, family socioeconomic and health wellness needs, and researchers and participants while being able to make specific design decisions from this organized view. Finally, we provide design implications and guidance for the development of JITAI mobile interventions for BIPOC families.