Hyper-relational facts, which consist of a primary triple (head entity, relation, tail entity) and auxiliary attribute-value pairs, are widely present in real-world Knowledge Graphs (KGs). Link Prediction on Hyper-relational Facts (LPHFs) is to predict a missing element in a hyper-relational fact, which helps populate and enrich KGs. However, existing LPHFs studies usually require an amount of high-quality data. They overlook few-shot relations, which have limited instances, yet are common in real-world scenarios. Thus, we introduce a new task, Few-Shot Link Prediction on Hyper-relational Facts (FSLPHFs). It aims to predict a missing entity in a hyper-relational fact with limited support instances. To tackle FSLPHFs, we propose MetaRH, a model that learns Meta Relational information in Hyper-relational facts. MetaRH comprises three modules: relation learning, support-specific adjustment, and query inference. By capturing meta relational information from limited support instances, MetaRH can accurately predict the missing entity in a query. As there is no existing dataset available for this new task, we construct three datasets to validate the effectiveness of MetaRH. Experimental results on these datasets demonstrate that MetaRH significantly outperforms existing representative models.