Resting-state fMRI (rs-fMRI) is an effective tool for quantifying functional connectivity (FC), which plays a crucial role in exploring various brain diseases. Due to the high dimensionality of fMRI data, FC is typically computed based on the region of interest (ROI), whose parcellation relies on a pre-defined atlas. However, utilizing the brain atlas poses several challenges including 1) subjective selection bias in choosing from various brain atlases, 2) parcellation of each subject's brain with the same atlas yet disregarding individual specificity; 3) lack of interaction between brain region parcellation and downstream ROI-based FC analysis. To address these limitations, we propose a novel randomizing strategy for generating brain function representation to facilitate neural disease diagnosis. Specifically, we randomly sample brain patches, thus avoiding ROI parcellations of the brain atlas. Then, we introduce a new brain function representation framework for the sampled patches. Each patch has its function description by referring to anchor patches, as well as the position description. Furthermore, we design an adaptive-selection-assisted Transformer network to optimize and integrate the function representations of all sampled patches within each brain for neural disease diagnosis. To validate our framework, we conduct extensive evaluations on three datasets, and the experimental results establish the effectiveness and generality of our proposed method, offering a promising avenue for advancing neural disease diagnosis beyond the confines of traditional atlas-based methods. Our code is available at https://github.com/mjliu2020/RandomFR.