As one of the fundamental modules in autonomous driving, online high-definition (HD) maps have attracted significant attention due to their cost-effectiveness and real-time capabilities. Since vehicles always cruise in highly dynamic environments, spatial displacement of onboard sensors inevitably causes shifts in real-time HD mapping results, and such instability poses fundamental challenges for downstream tasks. However, existing online map construction models tend to prioritize improving each frame's mapping accuracy, while the mapping stability has not yet been systematically studied. To fill this gap, this paper presents the first comprehensive benchmark for evaluating the temporal stability of online HD mapping models. We propose a multi-dimensional stability evaluation framework with novel metrics for Presence, Localization, and Shape Stability, integrated into a unified mean Average Stability (mAS) score. Extensive experiments on 42 models and variants show that accuracy (mAP) and stability (mAS) represent largely independent performance dimensions. We further analyze the impact of key model design choices on both criteria, identifying architectural and training factors that contribute to high accuracy, high stability, or both. To encourage broader focus on stability, we will release a public benchmark. Our work highlights the importance of treating temporal stability as a core evaluation criterion alongside accuracy, advancing the development of more reliable autonomous driving systems. The benchmark toolkit, code, and models will be available at https://stablehdmap.github.io/.