This study explores how socio-cultural dynamics influence student engagement with Generative AI technology in Nigerian higher education, using activity theory as theoretical underpinning. By examining the roles of community norms, technological accessibility, and educational objectives, the research identifies critical factors that impact the adoption and utilisation of GenAI. We employ quantitative analysis to analyse 899 survey responses from students across seventeen (17) Nigerian universities to derive interesting insights. Findings reveal that the ease of use of GenAI tools and their alignment with educational goals enhance student engagement. Conversely, regular need for technical support negatively affect engagement, suggesting underlying technological issues. These insights provide actionable recommendations for educators, administrators, and policymakers, emphasising the importance of user-friendly GenAI tools, comprehensive training programs, and robust support systems. This study contributes to the understanding of technology adoption in culturally diverse educational settings and offers strategies to improve educational practices and outcomes both in Nigerian higher education and also potentially in other African (developing countries), where similar socio-cultural dynamics might influence technology integration and educational advancements.