Increasing demand for power-efficient, high-performance computing has spurred a growing number and diversity of hardware accelerators in mobile and server Systems on Chip (SoCs). This paper makes the case that the co-design of the accelerator microarchitecture with the system in which it belongs is critical to balanced, efficient accelerator microarchitectures. We find that data movement and coherence management for accelerators are significant yet often unaccounted components of total accelerator runtime, resulting in misleading performance predictions and inefficient accelerator designs. To explore the design space of accelerator-system co-design, we develop gem5-Aladdin, an SoC simulator that captures dynamic interactions between accelerators and the SoC platform, and validate it to within 6% against real hardware. Our co-design studies show that the optimal energy-delay-product (EDP) of an accelerator microarchitecture can improve by up to 7.4× when system-level effects are considered compared to optimizing accelerators in isolation.