Simultaneous localization and mapping (SLAM) plays an important role in the state estimation of mobile robots. Most popular LiDAR SLAM (L-SLAM) methods extract feature points only from the geometric structure of the environment, which can result in inaccurate localization in degenerated scenarios. In this article, we present a novel framework for LiDAR intensity gradient enhanced tightly coupled LiDAR-inertial odometry (IGE-LIO). The framework proposes a novel LiDAR intensity gradient-based feature extraction approach for accurate pose estimation, overcoming the challenges faced by L-SLAM in degenerated environments. After computing the intensity gradient of each LiDAR point, we dynamically extract intensity edge points (IEPs) from texture information. In addition, we extract geometric planar points (GPPs) and geometric edge points (GEPs) based on geometric information. Then, the error analysis is performed on each type of feature points, and the weighting functions are designed to correct measurement noise and mitigate biases introduced by the additional uncertainty in feature extraction. Subsequently, an iterative extended Kalman filter (IEKF) framework is constructed by combining residuals from point-to-plane and point-to-edge associations. Finally, extensive experiments are conducted in indoor, outdoor, and LiDAR degenerated scenarios. The results demonstrate the significantly improved robustness and accuracy of our proposed method compared with the existing geometric-only methods, especially in LiDAR degenerated scenarios.