In random-effects meta-analysis the between-study variance ([tau][superscript 2]) has a key role in assessing heterogeneity of study-level estimates and combining them to estimate an overall effect. For odds ratios the most common methods suffer from bias in estimating [tau][superscript 2] and the overall effect and produce confidence intervals with below-nominal coverage. An improved approximation to the moments of Cochran's "Q" statistic, suggested by Kulinskaya and Dollinger (KD), yields new point and interval estimators of [tau][superscript 2] and of the overall log-odds-ratio. Another, simpler approach (SSW) uses weights based only on study-level sample sizes to estimate the overall effect. In extensive simulations we compare our proposed estimators with established point and interval estimators for [tau][superscript 2] and point and interval estimators for the overall log-odds-ratio (including the Hartung-Knapp-Sidik-Jonkman interval). Additional simulations included three estimators based on generalized linear mixed models and the Mantel-Haenszel fixed-effect estimator. Results of our simulations show that no single point estimator of [tau][superscript 2] can be recommended exclusively, but Mandel-Paule and KD provide better choices for small and large numbers of studies, respectively. The KD estimator provides reliable coverage of [tau][superscript 2]. Inverse-variance-weighted estimators of the overall effect are substantially biased, as are the Mantel-Haenszel odds ratio and the estimators from the generalized linear mixed models. The SSW estimator of the overall effect and a related confidence interval provide reliable point and interval estimation of the overall log-odds-ratio.