Cold-start issues have been more and more challenging for providing accurate recommendations with the fast increase of users and items. Most existing approaches attempt to solve the intractable problems via content-aware recommendations based on auxiliary information and/or cross-domain recommendations with transfer learning. Their performances are often constrained by the extremely sparse user-item interactions, unavailable side information, or very limited domain-shared users. Recently, meta-learners with meta-augmentation by adding noises to labels have been proven to be effective to avoid overfitting and shown good performance on new tasks. Motivated by the idea of meta-augmentation, in this paper, by treating a user's preference over items as a task, we propose a so-called Diverse Preference Augmentation framework with multiple source domains based on meta-learning (referred to as MetaDPA) to i) generate diverse ratings in a new domain of interest (known as target domain) to handle overfitting on the case of sparse interactions, and to ii) learn a preference model in the target domain via a meta-learning scheme to alleviate cold-start issues. Specifically, we first conduct multi-source domain adaptation by dual conditional variational autoencoders and impose a Multi-domain InfoMax (MDI) constraint on the latent representations to learn domain-shared and domain-specific preference properties. To avoid overfitting, we add a Mutually-Exclusive (ME) constraint on the output of decoders to generate diverse ratings given content data. Finally, these generated diverse ratings and the original ratings are introduced into the meta-training procedure to learn a preference meta-learner, which produces good generalization ability on cold-start recommendation tasks. Experiments on real-world datasets show our proposed MetaDPA clearly outperforms the current state-of-the-art baselines.