Large language models (LLMs) are increasingly used as proposal generators for evolutionary robot design, yet most loops remain memoryless: simulator results shape the next population but are not preserved as reusable design knowledge. We present Auto-Robotist, a self-evolving LLM agent that distills morphology-search traces into an explicit natural-language skill library. Each skill stores a structural archetype, evidence-grounded positive and negative rules, and the evaluated designs that support them, making design memory inspectable rather than implicit in a population. During search, the agent retrieves skills to condition LLM edits of elite bodies while retaining a Genetic Algorithm (GA) mutation path for exploration; after evaluation, it updates the library through Add, Diagnose, and Merge. Across seven EvoGym tasks spanning locomotion, traversal, and object interaction, Auto-Robotist improves cold-start 5×5 search and transfers learned skills to 10×10 design spaces, where reference-conditioned transfer outperforms or matches GA on every task when averaged across seeds. These results suggest that LLM agents can convert expensive physical evaluations into reusable, auditable design principles. Our code is publicly available at https://github.com/wangyf9/Auto-Robotist.