Rapid advancements in artificial intelligence technology have created novel challenges in personal data protection that existing legal frameworks struggle to address effectively. Using qualitative content analysis of scientific and legal documents published between 2018 and 2025, this research identifies and examines deficiencies in current legal systems when confronted with advanced AI capabilities. The findings reveal that challenges such as the "black box" problem of AI algorithms, conflict between data minimization principles and AI's need for extensive data, inefficiencies in informed consent mechanisms, gaps in regulatory coverage of foundation models, and lack of international regulatory harmonization are among the most significant shortcomings. This research proposes approaches for redefining key data rights concepts such as informed consent, the right to be forgotten, and transparency, while presenting a comprehensive framework consisting of fundamental principles, implementation mechanisms, and legal and technical solutions. The main innovation of this research is the introduction of a "distributed accountability" model that allocates responsibility among all actors in the AI ecosystem. This framework establishes a balanced approach to AI regulation by maintaining equilibrium between protecting fundamental individual rights and promoting innovation.