A rare disease (RD) affects fewer than five in 10,000 individuals, but collectively impacts around 300 million people worldwide and poses challenges for timely diagnosis due to limited data and complex symptom profiles. While advances in AI show promise for improving RD diagnosis, current system design might overlook stakeholder values such as inclusiveness and transparency. This gap is critical, as individuals with RDs frequently endure prolonged diagnostic journeys and unmet care needs. Participatory approaches can help unfold values, but are difficult to implement with geographically dispersed and hard-to-reach communities in the context of RD. In this paper, we argue that remote participation can enable value elicitation to inform the design of AI-supported RD diagnostic tools. We introduce an approach grounded in group-based online activities and reflect on its development in collaboration with RD care experts. Our findings highlight methodological reflections for remote research in sensitive contexts, including opportunities and limitations in recruiting diverse participants, leveraging elicited values to inform design, building a shared understanding of AI in RD diagnostics, and supporting the researcher–participant relationship.