I have a Sony Bravia KDL-40V5100 that won't turn on. Plug in, solid green light, power on, no click, no display or audio. Turn off, click, solid green light returns. Replace power supply board?

Fault at its main digital board. Short circuited component/s at it. Contact any service technician; or the authorized service center. If you wish to get some details; check the site linked here. Pull up older posts. http://electro-medical.blogspot.com/
In most cases, the main board might have to be replaced. If so, a software [firmware] update too might be necessary.
Before going further; try a hard reset first. If it does not help you, replace the main digital board. There are two methods to reset TVs. The first is simple to do; a Hard Reset. The other is somewhat skilled; is a Factory Reset (System Reset). To make it a hard reset, just unplug the TV from AC mains wall socket, and re-plug it back after 15 minutes or so. If you do this by overnight is better. This procedure will make it a hard reset. If you want to make it a system reset [Factory Reset]; you have to enter its service mode option, and select the factory reset option. If you wish to get some details; visit the site linked here.