We propose an accurate and robust initialization approach for stereo visual-inertial SLAM systems. Unlike the current state-of-the-art method, which heavily relies on the accuracy of a pure visual SLAM system to estimate inertial variables without updating camera poses, potentially compromising accuracy and robustness, our approach offers a different solution. We realize the crucial impact of precise gyroscope bias estimation on rotation accuracy. This, in turn, affects trajectory accuracy due to the accumulation of translation errors. To address this, we first independently estimate the gyroscope bias and use it to formulate a maximum a posteriori problem for further refinement. After this refinement, we proceed to update the rotation estimation by performing IMU integration with gyroscope bias removed from gyroscope measurements. We then leverage robust and accurate rotation estimates to enhance translation estimation via 3-DoF bundle adjustment. Moreover, we introduce a novel approach for determining the success of the initialization by evaluating the residual of the normal epipolar constraint. Extensive evaluations on the EuRoC dataset illustrate that our method excels in accuracy and robustness. It outperforms ORB-SLAM3, the current leading stereo visual-inertial initialization method, in terms of absolute trajectory error and relative rotation error, while maintaining competitive computational speed. Notably, even with 5 keyframes for initialization, our method consistently surpasses the state-of-the-art approach using 10 keyframes in rotation accuracy. The open source code is available at https://github.com/ApdowJN/Stereo-NEC.git.