Local differential privacy (LDP) has become a widely accepted framework for privacy-preserving data collection. In LDP, many protocols rely on hash functions to implement user-side encoding and perturbation. However, the security and privacy implications of hash function selection have not been previously investigated. In this paper, we expose that the hash functions may act as a source of unfairness in LDP protocols. We show that although users operate under the same protocol and privacy budget, differences in hash functions can lead to significant disparities in vulnerability to inference and poisoning attacks. To mitigate hash-induced unfairness, we propose Fair-OLH (F-OLH), a variant of OLH that enforces an entropy-based fairness constraint on hash function selection. Experiments show that F-OLH is effective in mitigating hash-induced unfairness under acceptable time overheads.