We consider the problem of user-adaptive 3D gaze estimation. The performance of person-independent gaze estimation is limited due to interpersonal anatomical differences. Our goal is to provide a personalized gaze estimation model specifically adapted to a target user. Previous work on user-adaptive gaze estimation requires some labeled images of the target person data to fine-tune the model at test time. However, this can be unrealistic in real-world applications, since it is cumbersome for an end-user to provide labeled images. In addition, previous work requires the training data to have both gaze labels and person IDs. This data requirement makes it infeasible to use some of the available data. To tackle these challenges, this paper proposes a new problem called efficient label-free user adaptation in gaze estimation. Our model only needs a few unlabeled images of a target user for the model adaptation. During offline training, we have some labeled source data without person IDs and some unlabeled person-specific data. Our proposed method uses a meta-learning approach to learn how to adapt to a new user with only a few unlabeled images. Our key technical innovation is to use a generalization bound from domain adaptation to define the loss function in meta-learning, so that our method can effectively make use of both the labeled source data and the unlabeled person-specific data during training. Extensive experiments validate the effectiveness of our method on several challenging benchmarks.