Lightning rod structures are susceptible to wind loads due to their high slenderness ratio, high flexibility, and light weight. The wind-induced dynamic response of a lightning rod is critical for structural safety and reliability. The traditional methods for this response, including observation and simulation, focus on structural health monitoring (SHM), wind tunnel tests (WTTs), or fluid–structure interaction (FSI) simulations. However, all these approaches require considerable financial or computational investment. Additionally, problems such as data loss or data anomalies in the sensor monitoring process often occur during SHM or WTTs. This paper proposes an algorithm based on a long short-term memory (LSTM) network to predict the wind-induced dynamic response and to solve the problem of data link fracture caused by abnormal sensor data transmission or wind-induced damage to lightning rod structures under different wind speeds. The effectiveness and applicability of the proposed framework are demonstrated using actual monitoring data. Root-mean-squared error (RMSE), determination of coefficient (R2), variance accounted for (VAF), and the refined Willmott index (RWI) are employed as performance assessment indices for the proposed network model. At the same time, the random forest algorithm is adopted to analyze the correlation between the data of the different measurement points on the lightning rod structure. The results show that the LSTM method proposed in this paper has a high accuracy for the prediction of “missing” strain data during lightning rod strain monitoring under wind speeds of 15.81~31.62 m/s. Even under the extreme wind speed of 31.62 m/s, the values of RMSE, MAE, R2, RWI and VAF are 0.24053, 0.18213, 0.94539, 0.88172 and 0.94444, respectively, which are within the acceptable range. Using the data feature importance analysis function, it is found that the predicted strain data of the measurement point on the top part of the lightning rod structure are closely related to the test strain data of the two adjacent sections of the structure, and the effect of the test strain data of the measurement points that are far from the predicted measurement point can be ignored.