Universities are at a crossroads as Generative AI (GenAI) transforms the landscape of academic innovation and competition. This research addresses the strategic dilemma facing U.S. universities: how to protect and maximize the value of AI-assisted discoveries while advancing their core missions. Through a qualitative methodology, combining literature review, expert interviews, and scenario-based analysis, the study examines both patenting and non-patenting pathways for GenAI outputs. Findings reveal that while patents can offer exclusivity and commercial potential, universities often benefit from integrating alternative strategies such as industry partnerships, capacity building, and data deployment. The proposed decision framework enables institutions to align intellectual property management with evolving technological, regulatory, and mission-driven priorities. Ultimately, this work provides actionable guidance for universities seeking to navigate the complexities of GenAI innovation, ensuring that academic research remains a driving force in technological progress.