Generating free-text rationales is a promising step towards explainable NLP, yet evaluating such rationales remains a challenge. Existing metrics have mostly focused on measuring the association between the rationale and a given label. We argue that an ideal metric should focus on the new information uniquely provided in the rationale that is otherwise not provided in the input or the label. We investigate this research problem from an information-theoretic perspective using conditional V-information (Hewitt et al., 2021). More concretely, we propose a metric called REV (Rationale Evaluation with conditional V-information), to quantify the amount of new, label-relevant information in a rationale beyond the information already available in the input or the label. Experiments across four benchmarks with reasoning tasks, including chain-of-thought, demonstrate the effectiveness of REV in evaluating rationale-label pairs, compared to existing metrics. We further demonstrate REV is consistent with human judgments on rationale evaluations and provides more sensitive measurements of new information in free-text rationales. When used alongside traditional performance metrics, REV provides deeper insights into models' reasoning and prediction processes.