Severe image degradation under low-light nighttime conditions constitutes a core bottleneck preventing all-day applications for UAV-based single object tracking. Existing image enhancement methods often struggle to distinguish between target and background regions, which can easily lead to amplified background noise or compromise target features. To overcome this limitation, we propose TAE, a target-aware low-light enhancement framework tailored for nighttime object tracking. Guided explicitly by weak supervisory signals from tracking bounding boxes, the framework performs region-aware enhancement to ensure operations focus on the target area. It further adopts an adaptive RGB multi-curve fusion mechanism to achieve refined modeling and adaptive adjustment across different regions. To facilitate research in this domain, we also contribute DarkSOT, a new benchmark for nighttime UAV tracking, comprising 268 sequences across 9 target categories. Experimental results on the DarkSOT and UAVDark135 demonstrate that TAE significantly improves tracking performance in low-light nighttime scenarios, exhibiting strong robustness and generalization. The DarkSOT dataset is available at https://github.com/Fu0511/DarkSOT-Dataset.