Background and Objective: Biomedical Named Entity Recognition (BioNER) is a foundational task in medical informatics, crucial for downstream applications like drug discovery and clinical trial matching. However, adapting general-domain Large Language Models (LLMs) to this task is often hampered by their lack of domain-specific knowledge and the performance degradation caused by low-quality training data. To address these challenges, we introduce BioSelectTune, a highly efficient, data-centric framework for fine-tuning LLMs that prioritizes data quality over quantity. Methods and Results: BioSelectTune reformulates BioNER as a structured JSON generation task and leverages our novel Hybrid Superfiltering strategy, a weak-to-strong data curation method that uses a homologous weak model to distill a compact, high-impact training dataset. Conclusions: Through extensive experiments, we demonstrate that BioSelectTune achieves state-of-the-art (SOTA) performance across multiple BioNER benchmarks. Notably, our model, trained on only 50% of the curated positive data, not only surpasses the fully-trained baseline but also outperforms powerful domain-specialized models like BioMedBERT.