As hospitals have experienced a surge of Covid-19 patients, investigators of Covid-19 treatment trials face a difficult problem: when an institution has more eligible and interested patients than trial slots, who should be enrolled? Defining a clear strategy for selecting participants for "high-demand" Covid-19 treatment trials is important to avoid ad hoc and potentially biased decision-making by local investigators, which could inadvertently compromise a trial's social value, participants' interests, or fairness. In this article, we propose a set of ethical criteria for evaluating participant-selection strategies for such trials. We argue that the pandemic context-in particular, great urgency to develop safe and effective treatments, uncertainty surrounding Covid-19, and strain on the health care system that limits the time and effort available for trial enrollment-favors participant-selection strategies that optimize the ease of enrollment and, ideally, social value. A lottery and, where possible, a weighted lottery have important advantages in these respects.