Protecting Personally Identifiable Information (PII), such as names, is a critical requirement in learning technologies to safeguard student and teacher privacy and maintain trust. Accurate PII detection is an essential step toward anonymizing sensitive information while preserving the utility of educational data. Motivated by recent advancements in artificial intelligence, our study investigates the GPT-4o-mini model as a cost-effective and efficient solution for PII detection tasks. We explore both prompting and fine-tuning approaches and compare GPT-4o-mini's performance against established frameworks, including Microsoft Presidio and Azure AI Language. Our evaluation on two public datasets, CRAPII and TSCC, demonstrates that the fine-tuned GPT-4o-mini model achieves superior performance, with a recall of 0.9589 on CRAPII. Additionally, fine-tuned GPT-4o-mini significantly improves precision scores (a threefold increase) while reducing computational costs to nearly one-tenth of those associated with Azure AI Language. Furthermore, our bias analysis reveals that the fine-tuned GPT-4o-mini model consistently delivers accurate results across diverse cultural backgrounds and genders. The generalizability analysis using the TSCC dataset further highlights its robustness, achieving a recall of 0.9895 with minimal additional training data from TSCC. These results emphasize the potential of fine-tuned GPT-4o-mini as an accurate and cost-effective tool for PII detection in educational data. It offers robust privacy protection while preserving the data's utility for research and pedagogical analysis. Our code is available on GitHub: https://github.com/AnonJD/PrivacyAI