This study evaluates the performance of an active reconfigurable intelligent surface (ARIS)-assisted non-orthogonal multiple access (NOMA) system employing low-precision analog-to-digital converters (ADCs). Analytical approximations for the outage probability (OP) are derived, considering residual hardware impairments (RHIs) and imperfect successive interference cancellation (ipSIC). Additionally, we analyze the asymptotic OP, system throughput, and diversity order at high signal-to-noise ratios (SNRs). Simulation results demonstrate that the proposed quantized ARIS-NOMA system outperforms its passive counterpart (PRIS-NOMA), achieving lower OP and higher throughput with reduced transmit power requirements and fewer reflecting elements. Moreover, the outage performance of both quantized ARIS-NOMA and PRIS-NOMA systems demonstrates significant improvement as the number of reflecting elements increases. The negative impacts of low-precision ADCs can be effectively mitigated by optimizing transmit power and scaling the number of reflecting elements.