Endoscopic reconstruction and rendering technology are crucial for minimally invasive diagnosis and treatment. Due to physical constraints such as limited light source placement and narrow operational space, endoscopic scenes often contain some dark regions that compromise clinical observation and diagnostic accuracy. To address this issue, we propose AdaEndoGS for endoscopic 3D reconstruction with physically consistent dark region illumination enhancement. Specifically, built upon 3D Gaussian Splatting, AdaEndoGS enriches each Gaussian with surface attributes including normal, roughness, and reflectance, thereby constructing an illuminatable 3D scene representation. AdaEndoGS automatically identifies the dark region via ray tracing, leveraging Gaussian opacity, surface normal, and viewpoint information to adaptively plan the placement of a virtual light source. Supervised by a carefully designed loss function incorporating multiple illumination-related terms, AdaEndoGS optimizes light source intensity and attenuation parameters to generate harmonious illumination enhancement in the dark regions, while avoiding overexposure in originally bright areas. We comprehensively evaluate our method on the public dataset C3VD and a novel endoscopy simulation dataset created by us in Unity3D, which includes paired low-lit and well-lit images to facilitate quantitative evaluation. Experimental results show that AdaEndoGS more accurately simulates light-matter interactions compared to existing methods. It significantly improves visual quality and enhances detail visibility, offering an effective technical solution for advancing endoscopic image-based reconstruction and rendering. Project repository: https://webstermorton.github.io/adaendogs-website.