Recently, memristors received considerable attention in various applications. Even some of the main drawbacks of resistive memory devices (RRAM), such as variability, have become attractive features for hardware security in the form of a Physically Unclonable Function (PUF). Although several RRAM-based PUFs have appeared in the literature, they still suffer from some issues related to reliability, reconfigurability, and extensive integration cost. This paper presents a novel lightweight reconfigurable RRAM-based PUF (LRR-PUF) wherein multiple RRAM cells, connected to the same bit line and same transistor (1T4R), are used to generate a single bit response. The pulse programming method used is also innovative: 1) it allows for a power-efficient implementation, and 2) it exploits variations in the number of pulses needed to switch the RRAM cell as the primary entropy source of the PUF. The main feature of the proposed PUF is its integration with any RRAM architecture at almost no additional cost. Through extensive simulations, including the impact of temperature and voltage variations along with statistical characterization, we demonstrate that the LRR-PUF exhibits such attractive properties that are lacking or poorly achieved in other previously proposed RRAM based PUFs, including high reliability (almost 100%), which is critical for cryptographic key generation, reconfigurability, uniqueness, cost, and efficiency. Furthermore, the design successfully passes relevant NIST tests for randomness.