Large Language Models (LLMs) show promise for knowledge graph (KG) reasoning and link prediction, but often fail to exploit graph structure, exhibit greedy generation and hallucination, and require costly updates; meanwhile, traditional KG completion methods can be less expressive and generalize poorly. This paper proposes Random-LLM, a stochastic prompting architecture for LLM-based link prediction without fine-tuning that injects structural and inferential diversity. It combines random walks on the KG and its moral graph to supply structural cues, and conditions on both best and worst prior generations to reduce overfitting and greedy selection. On CoDeX-M, UMLS, and WN18RR with FLAN-T5-Base, Random-LLM improves Hit@1 and precision by around 20% and reaches up to 70% accuracy, matching or surpassing several KG embedding baselines, highlighting the value of randomized, structure-aware prompting for KG reasoning.