With the proliferation of low-cost mobile edge devices, security and reliability have become crucial for mobile edge computing networks, especially for high-stakes applications. To facilitate it, Radio Frequency Fingerprint Identification (RFFI) has emerged as a promising physical layer security paradigm, offering a non-cryptographic and lightweight solution. However, most existing Deep Learning (DL) based RFFI methods operate under a closed-set assumption, limiting their ability to recognize devices not seen during training and posing a risk of misclassification. Addressing the open-set RFFI problem is critical for real-world deployments, where the system must handle both known and unknown devices, ensuring robust security in dynamic environments. In this paper, we propose OpenRFI, a novel test-time fine-tuning-based RFFI framework, consisting of two sequential stages: pre-training and test-time fine-tuning. During the pre-training stage, we design a data augmentation module, a feature extraction module, and an efficient hybrid loss function to minimize intra-class feature distances and tighten decision boundaries, enhancing the model's ability to distinguish between different classes. In the test-time fine-tuning stage, we introduce a fine-tuning dataset construction module and a full-parameter fine-tuning module to dynamically adapt to the test environment and capture information from unknown samples, further improving open-set recognition. We theoretically establish the performance boundary of the fine-tuning dataset construction method, providing insights into its robustness and scalability. Extensive numerical results based on an open source dataset demonstrate the effectiveness of the proposed OpenRFI framework in comparison with existing baselines.