Accurate knowledge of the rotor position is essential for the control of brushless DC motors (BLDCM). Any deviation in this identification can cause fluctuations in motor current and torque, increase noise, and lead to reduced motor efficiency. This paper focused on a BLDCM equipped with a three-phase binary Hall sensor. Based on the principle of minimum deviation, this paper estimated the relative installation offset between the Hall sensors. It also provided a clear method for ideal phase commutation position recognition and eliminated the Hall sensor installation position deviation. The proposed pre-calibration method identified and eliminated the offset of the permanent magnet poles, the delay time caused by the Hall signal conditioning circuit, and the offset of the sensor signal identification due to armature response under different loads. Based on the pre-calibration results, a correction strategy for correcting the rotor position information of BLDCMs was proposed. This paper presented a self-adaptive position information prediction algorithm based on the Sage–Husa method. This filters out rotor position information deviations that are not eliminated in pre-calibration. Experimental results on a hydrogen circulation pump motor showed that, after the pre-calibration method was adopted, the Mean Square Error (MSE) of motor speed fluctuations decreased by 92.0%, motor vibration was significantly reduced, average phase current decreased by 62.8%, and the efficiency of the hydrogen circulation pump system was significantly improved. Compared to the traditional KF prediction algorithm, the Sage–Husa adaptive position information prediction algorithm reduced the speed fluctuation during the uniform speed operation stage and speed adjustment stage, the speed curve overshoot, and the commutation time deviation throughout the process by 44.8%, 56.0%, 54.9%, and 14.7%, respectively. This indicates a higher disturbance rejection ability and a more accurate and stable prediction of the commutation moment.