The channel fading generated by seawater absorption and scattering, oceanic turbulence, and pointing error will severely reduce the communication quality of an Underwater Wireless Optical Communication (UWOC) system. Many scholars directly transplanted the weak atmospheric turbulence model to depict the oceanic turbulence statistics, and this had been proven to be incorrect by a series of laboratory measurements and associated data-fitting tests. So, it is obviously of great significance to study and evaluate the effects of a composite fading channel on the key performance of the UWOC system, especially with a proper weak oceanic turbulence model.In this paper, the Generalized Gamma Distribution (GGD) verified by a series of experimental tests was selected to characterize the weak oceanic turbulence. Then, a new hybrid fading channel model was proposed to more reasonably simulate the communication environment in the ocean, which had integrated the GGD weak turbulence, the zero/nonzero boresight pointing error, the implicit path loss and multipath propagation effect characterized by Fading Free Impulse Response (FFIR). Next, the mathematical expressions of the Probability Density Function (PDF) considering GGD weak turbulence and zero/nonzero boresight pointing errors were derived using higher transcendental Meijer-G and Whittaker functions. Subsequently, based on this, the closed-form expressions of the average Bit Error Rate (BER) were derived for the serial-relayed UWOC systems with both zero and nonzero boresight pointing errors, respectively. Finally, the accuracy and rationality of the derived closed-form formulas for the average bit error rate of the relaying UWOC system derived above were verified by some Monte Carlo numerical simulations; meanwhile, the influences of different key parameters on the system BERs were also investigated.The results show that the introduction of serial relaying nodes can effectively improve the end-to-end BER performance of the UWOC systems in a long-distance communication environment. With the increase of the relaying nodes number, the system's BER decreases rapidly at the same transmission power, indicating that the serial-relayed scheme dramatically improves the performance. For instance, if the end-to-end distance is fixed as 45 m and the target BER is set to be 10^-3, under the zero boresight pointing error condition, the required node transmission power will be reduced to 24 dBm, 12 dBm, and 6 dBm, respectively, when the relaying node number is assumed to be 1 to 3 individually. Similar to the working situation of zero boresight pointing error, when there is a non-zero boresight pointing error, the serial relaying node still effectively improves the system's BER performance. Unfortunately, the delayed spread expansion of the FFIR caused by the increase of the initial divergence angle of light source, that is, the so-called Inter-symbol Interference (ISI), will seriously degrade this performance improvement. For example, when the initial divergence angle is increased from 0.01° to 3°, the transmission power needs to be increased in general by about 10~15 dBm to obtain the same BER compared with the original transmission one. In addition, the jitter standard deviation and the initial boresight displacement associated with the pointing error will also significantly impact the BER performance of the relaying UWOC systems. For instance, if the jitter standard deviation is enhanced from 10 cm to 20 cm, the node transmission power needs to be increased by nearly 20~25 dBm to achieve the same BER value; meanwhile, when the initial boresight displacement increases from 5 cm to 10 cm, the transmission power needs to be added by about 5 dBm to reach the same BER target. Therefore, when one considers the BER evaluation of the serial-relayed UWOC systems, it is necessary to consider the impacts of the initial divergence angle, the jitter standard deviation, and the initial boresight displacement on performance thoroughly.The analytical results of this paper can provide calculation support for analyzing the BER performance of the relaying UWOC systems.