Societal change is often driven by shifts in public opinion. As citizens evolve in their norms, beliefs, and values, public policies change too. While traditional opinion polling and surveys can outline the broad strokes of whether public opinion on a particular topic is changing, they usually cannot capture the full multi-dimensional richness and diversity of opinion present in a large heterogeneous population. However, an increasing fraction of public discourse about public policy issues is now occurring on online platforms, which presents an opportunity to measure public opinion change at a qualitatively different scale of resolution and context. In this paper, we present a conceptual model of observed opinion change on online platforms and apply it to study public discourse on Universal Basic Income (UBI) on Reddit throughout its history. UBI is a periodic, no-strings-attached cash payment given to every citizen of a population. We study UBI as it is a clearly-defined policy proposal that has recently experienced a surge of interest through trends like automation and events like the COVID-19 pandemic. We find that overall stance towards UBI on Reddit significantly declined until mid-2019, when this historical trend suddenly reversed and Reddit became substantially more supportive. Using our model, we find the most significant drivers of this overall stance change were shifts within different user cohorts, within communities that represented similar affluence levels, and within communities that represented similar partisan leanings. Our method identifies nuanced social drivers of opinion change in the large-scale public discourse that now regularly occurs online, and could be applied to a broad set of other important issues and policies.