State estimation in complex illumination environments based on conventional visual-inertial odometry is a challenging task due to the severe visual degradation of the visual camera. The thermal infrared camera is capable of all-day time and is less affected by illumination variation. However, most existing visual data association algorithms are incompatible because the thermal infrared data contains large noise and low contrast. Motivated by the phenomenon that thermal radiation varies most significantly at the edges of objects, the study proposes an ETIO, which is the first edge-based monocular thermal-inertial odometry for robust localization in visually degraded environments. Instead of the raw image, we utilize the binarized image from edge extraction for pose estimation to overcome the poor thermal infrared image quality. Then, an adaptive feature tracking strategy ADT-KLT is developed for robust data association based on limited edge information and its distance distribution. Finally, a pose graph optimization performs real-time estimation over a sliding window of recent states by combining IMU pre-integration with reprojection error of all edge feature observations. We evaluated the performance of the proposed system on public datasets and real-world experiments and compared it against state-of-the-art methods. The proposed ETIO was verified with the ability to enable accurate and robust localization all-day time.