The increasing complexity of real-world engineering problems, ranging from manufacturing scheduling to resource optimization in smart grids, has driven demand for adaptive and high-performing heuristic methods. Automatic Heuristic Design (AHD) and neural-enhanced metaheuristics have shown promise in automating strategy development, but often suffer from limited flexibility and scalability due to static operator libraries or high retraining costs. Recently, Large Language Models (LLMs) have emerged as a powerful alternative for exploring and evolving heuristics through natural language and program synthesis. This paper proposes a novel LLM-based memetic framework that synergizes LLM-driven exploration with domain-specific local refinement and memory-aware reflection, enabling a dynamic balance between heuristic creativity and effectiveness. In the experiments, the developed framework outperforms other LLM-based state-of-the-art approaches across the designed AGV-drone scheduling scenario and two benchmark combinatorial problems. The findings suggest that LLMs can serve not only as general-purpose optimizers but also as interpretable heuristic generators that adapt efficiently to complex and heterogeneous domains.