To address an important gap in creating children's stories for vocabulary enrichment, we investigate the automatic evaluation of how well stories convey the semantics of target vocabulary words, a task with substantial implications for generating educational content. We motivate this task, which we call measuring contextual informativeness in children's stories, and provide a formal task definition as well as a dataset for the task. We further propose a method for automating the task using a large language model (LLM). Our experiments show that our approach reaches a Spearman correlation of 0.4983 with human judgments of informativeness, while the strongest baseline only obtains a correlation of 0.3534. An additional analysis shows that the LLM-based approach is able to generalize to measuring contextual informativeness in adult-directed text, on which it also outperforms all baselines.