Large language models (LLMs) are increasingly used in computational social science, both to analyze human-generated web content and to act as human surrogates capable of answering substantive research questions. As these uses expand, a critical methodological question arises: how stable and reliable are the answers LLMs produce? We examine this question through the lens of stance mining on gun control and gun rights, a mature and widely used NLP task applied to one of the most extensively studied and polarized political issues in the United States. Using a multi-platform dataset spanning diverse online communities, we evaluate the stability of LLM-predicted stance across model families, prompt designs, and conversational contexts. Despite strong expectations of robustness in this setting, we observe substantial variability in predicted stance across LLMs and prompts. To contextualize these findings, we compare model-induced variability with human variability by analyzing annotations from raters with differing political affiliations. We find that political priors and annotator disagreement persist even in this ostensibly straightforward classification task. As a useful resource, we release a stance dataset annotated by raters with diverse political affiliations. Our results show that outcomes in stance mining depend jointly on which LLM is used and which humans provide the inputs, underscoring dual sources of variability that have important implications for using LLMs as research instruments in computational social science.