The likely reason why your Honda Odyssey is not starting and has no clicking noise is that you have a bad starter or starter solenoid. When you turn the key, the battery supplies power to the starter, which in turn, fires up the engine. When the starter or starter solenoid goes bad, it fails to start and the noise you normally hear—the starter itself—doesn’t happen.
Depending on the model year, the cost to replace a bad starter or starter solenoid in a Honda Odyssey ranges from $400 to $800. The part itself isn’t all that expensive, but getting to it can be a pain based on the year of the vehicle.