Cross-target generalization is a known problem in stance detection (SD), where systems tend to perform poorly when exposed to targets unseen during training. Given that data annotation is expensive and time-consuming, finding ways to leverage abundant unlabeled in-domain data can offer great benefits. In this paper, we apply a weakly supervised framework to enhance cross-target generalization through synthetically annotated data. We focus on Twitter SD and show experimentally that integrating synthetic data is helpful for cross-target generalization, leading to significant improvements in performance, with gains in F1 scores ranging from +3.4 to +5.1.