In the field of domain adaptation (DA) on 3D object detection, most of the work is dedicated to unsupervised domain adaptation (UDA). Yet, without any target annotations, the performance gap between the UDA approaches and the fully-supervised approach is still noticeable, which is impractical for real-world applications. On the other hand, weakly-supervised domain adaptation (WDA) is an underexplored yet practical task that only requires few labeling effort on the target domain. To improve the DA performance in a cost-effective way, we propose a general weak labels guided self-training framework, WLST, designed for WDA on 3D object detection. By incorporating autolabeler, which can generate 3D pseudo labels from 2D bounding boxes, into the existing self-training pipeline, our method is able to generate more robust and consistent pseudo labels that would benefit the training process on the target domain. Extensive experiments demonstrate the effectiveness, robustness, and detector-agnosticism of our WLST framework. Notably, it outperforms previous state-of-the-art methods on all evaluation tasks.