Smartphone usage data can provide valuable insights for understanding interaction with technology and human behavior. However, collecting large-scale, in-the-wild smartphone usage logs is challenging due to high costs, privacy concerns, under representative user samples and biases like non-response that can skew results. These challenges call for exploring alternative approaches to obtain smartphone usage datasets. In this context, large language models (LLMs) such as Open AI's ChatGPT present a novel approach for synthetic smartphone usage data generation, addressing limitations of real-world data collection. We describe a case study on how four prompt strategies influenced the quality of generated smartphone usage data. We contribute with insights on prompt design and measures of data quality, reporting a prompting strategy comparison combining two factors, prompt level of detail (describing a user persona, describing the expected results characteristics) and seed data inclusion (with versus without an initial real usage example). Our findings suggest that using LLMs to generate structured and behaviorally plausible smartphone use datasets is feasible for some use cases, especially when using detailed prompts. Challenges remain in capturing diverse nuances of human behavioral patterns in a single synthetic dataset, and evaluating tradeoffs between data fidelity and diversity, suggesting the need for use-case-specific evaluation metrics and future research with more diverse seed data and different LLM models.