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After a large language model (LLM) is deployed on edge devices, it is desirable for these devices to learn from user-generated conversation data to generate user-specific and personalized responses in real-time. However, user-generated data usually contains sensitive and private information, and uploading such data to the cloud for annotation is not preferred if not prohibited. While it is possible to obtain annotation locally by directly asking users to provide preferred responses, such annotations have to be sparse to not affect user experience. In addition, the storage of edge devices is usually too limited to enable large-scale fine-tuning with full user-generated data. It remains an open question how to enable on-device LLM personalization, considering sparse annotation and limited on-device storage. In this paper, we propose a novel framework to select and store the most representative data online in a self-supervised way. Such data has a small memory footprint and allows infrequent requests of user annotations for further fine-tuning. To enhance fine-tuning quality, multiple semantically similar pairs of question texts and expected responses are generated using the LLM. Our experiments show that the proposed framework achieves the best user-specific content-generating capability (accuracy) and fine-tuning speed (performance) compared with vanilla baselines. To the best of our knowledge, this is the very first on-device LLM personalization framework.