Does anyone know how to replace a burned out backlight on a honeyw. RTH6300 thermostat?

I believe you will be buying a new thermostat.
No mention of it here.

http://customer.honeywell.com/techlit/pdf/69-0000s/69-1717.pdf

It probably takes a fair amount of effort to do so, but I'll bet it's not impossible. It will likely involve soldering. Look for CCFL or cold cathode fluorescent lamps and try to identify.

A flashlight would be much cheaper