Knowledge editing aims to update outdated information in Large Language Models (LLMs). A representative line of study is locate-then-edit methods, which typically employ causal tracing to identify the modules responsible for recalling factual knowledge about entities. However, we find these methods are often sensitive only to changes in the subject entity, leaving them less effective at adapting to changes in relations. This limitation results in poor editing locality, which can lead to the persistence of irrelevant or inaccurate facts, ultimately compromising the reliability of LLMs. We believe this issue arises from the insufficient precision of knowledge localization. To address this, we propose a Fine-grained Neuron-level Knowledge Editing (FiNE) method that enhances editing locality without affecting overall success rates. By precisely identifying and modifying specific neurons within feed-forward networks, FiNE significantly improves knowledge localization and editing. Quantitative experiments demonstrate that FiNE efficiently achieves better overall performance compared to existing techniques, providing new insights into the localization and modification of knowledge within LLMs.