Global Navigation Satellite System (GNSS) is indispensable in disaster monitoring, especially during extreme weather events such as extreme precipitation and typhoons. However, extreme precipitation can cause short-term anomalies in the Total Electron Content (TEC), leading to ionospheric errors that compromise GNSS accuracy and hinder effective disaster monitoring. Consequently, accurate determination of TEC is thus crucial for mitigating these errors and enhancing GNSS performance under such conditions. This paper focuses on the impact of extreme precipitation on TEC prediction accuracy and innovatively introduces specific humidity as an additional feature. This leads to the development of a Feature-Extended Long Short-Term Memory (LSTM) Model, designed to improve short-term ionospheric TEC prediction under challenging weather conditions. The prediction accuracy of the proposed model was evaluated using ionospheric TEC data, along with corresponding specific humidity, global geomagnetic index (Kp), and equatorial geomagnetic index (Dst) data. Experimental results demonstrate that the proposed Feature-Extended LSTM Model significantly improves prediction accuracy for TEC over 3–48-h forecast periods compared to both the Autoregressive Integrated Moving Average Model (ARIMA) model and the Non-Feature-Extended LSTM Model. Specifically, for a 48-h forecast period, the RMSE and MAE of the proposed model are improved by 42.88% and 48.29%, respectively, compared to the ARIMA model, and by 17.42% and 15.82%, respectively, compared to the Non-Feature-Extended LSTM Model.