As language model (LM) agents become increasingly capable and adopted in real-world applications, there is a growing need for scalable evaluation frameworks beyond costly, manually designed benchmarks. We propose information-theoretic evaluation based on empowerment, an information-theoretic measure of an agent's influence on future states through its actions. To handle the unique challenges of text-based environments, we introduce EELMA (Estimating Empowerment of Language Model Agents), an algorithm for approximating effective empowerment from multi-turn text interactions. We demonstrate EELMA on textual games and realistic web and tool-use environments, showing that empowerment strongly correlates with average task performance. We further analyze how empowerment varies across models, environment complexity, and agent configurations, and show that high-empowerment states and actions often mark pivotal moments for general capabilities. These results establish empowerment as a goal-agnostic metric that complements task-success measures for LM-agent evaluation.