High-precision ranging plays a crucial role in future 6G Integrated Sensing and Communication (ISAC) systems. To improve the ranging performance while maximizing the resource utilization efficiency, future 6G ISAC networks have to reuse data payload signals for both communication and sensing, whose inherent randomness may deteriorate the ranging performance. To address this issue, this paper investigates the power allocation (PA) design for an OFDM-based ISAC system under random signaling, aiming to reduce the ranging sidelobe level of both periodic and aperiodic auto-correlation functions (P-ACF and A-ACF) of the ISAC signal. Towards that end, we first derive the closed-form expressions of the average squared P-ACF and A-ACF, and then propose to minimize the expectation of the integrated sidelobe level (EISL) under arbitrary constellation mapping. We then rigorously prove that the uniform PA scheme achieves the global minimum of the EISL for both P-ACF and A-ACF. As a step further, we show that this scheme also minimizes the P-ACF sidelobe level at every lag. Moreover, we extend our analysis to the P-ACF case with frequency-domain zero-padding, which is a typical approach to improve the ranging resolution. We reveal that there exists a tradeoff between sidelobe level and mainlobe width, and propose a project gradient descent algorithm to seek a locally optimal PA scheme that reduces the EISL. Finally, we validate our theoretical findings through extensive simulation results, confirming the effectiveness of the proposed PA methods in reducing the ranging sidelobe level for random OFDM signals.