Alzheimer's disease prognosis is critical for early Mild Cognitive Impairment patients for timely treatment to improve the patient's quality of life. Whilst existing prognosis techniques demonstrate potential results, they are highly limited in terms of using a single modality. Most importantly, they fail in considering a key element for prognosis: not all features extracted at the current moment may contribute to the prognosis prediction several years later. To address the current drawbacks of the literature, we propose a novel hypergraph framework based on an information bottleneck strategy (HGIB). Firstly, our framework seeks to discriminate irrelevant information, and therefore, solely focus on harmonising relevant information for future MCI conversion prediction (e.g., two years later). Secondly, our model simultaneously accounts for multi-modal data based on imaging and non-imaging modalities. HGIB uses a hypergraph structure to represent the multi-modality data and accounts for various data modality types. Thirdly, the key of our model is based on a new optimisation scheme. It is based on modelling the principle of information bottleneck into loss functions that can be integrated into our hypergraph neural network. We demonstrate, through extensive experiments on ADNI, that our proposed HGIB framework outperforms existing state-of-the-art hypergraph neural networks for Alzheimer's disease prognosis. We showcase our model even under fewer labels. Finally, we further support the robustness and generalisation capabilities of our framework under both topological and feature perturbations.