Mediation analysis is widely used to study whether the effect of an independent variable on an outcome is transmitted through a mediator. Bayesian methods have become increasingly popular for mediation analysis. However, limited research has been done on formal Bayesian hypothesis testing of mediation. Although hypothesis testing using Bayes factor for a single path is readily available, how to integrate the Bayes factors of two paths (from input to mediator and from mediator to outcome) while incorporating prior beliefs on the two paths and/or mediation is under-studied. In the current study, we propose a general approach to Bayesian hypothesis testing of mediation. The proposed approach allows researchers to specify prior odds based on the substantive research context and can be used in mediation modeling with latent variables. The impact of prior odds specifications on Bayesian hypothesis test of mediation is demonstrated via both real and hypothetical data examples. Both R functions and a user-friendly R web app are provided for the implementation of the proposed approach. Our study can add to researchers' toolbox of mediation analysis and raise researchers' awareness of the importance of prior odds specifications in Bayesian hypothesis testing of mediation. [This is the online version of an article published in "Behavioral Research Methods."]