I've seen some use "shine on" to close a letter. What does this mean?
This is a rather old turn of phrase. It appears all over the King James Bible, both directly and indirectly. It roughly means "Don't hide from the world what a wonderful person you truly are."
Quite a lot of popular songs use the phrase. Off the top of my head, John Lennon's Instant Karma, Pink Floyd's Shine On You Crazy Diamond, and the old Gospel song This Little Light of Mine. All are rather influential songs in their places.
The correct meaning for "shine it on"dates back to 1931, when the county of Los Angeles Probation Department established its first forestry camp for at risk youths. This became the model for the federal CCC conservation camps. As part of their uniforms the kids wore hard hats with a silver metal brim, which they were expected to keep highly polished, ie. shiny. While wearing them on a work detail, if another kid attempted to irritate or provoke you usually a friend would counsel you to "shine it, or him, on" meaning to tilt your head down so that the would-be provocateur could only see his own reflection in the brim of your hat.
As result of the five juvenile deaths sustained during the 1955 Hacienda fire juvenile firefighting activities were sharply curtailed. The camps continued to operate but they now engaged in fire prevention such as brush clearance and not in direct fire suppression. I understand that at present there is one probation camp for older youth, Camp Routh, where the kids do fight fires. Don't know if they still wear those shiny helmets, though.
You, being a candle or another source of light, burning bright and shining light into an imaginary dark room, are wished by the person to keep defying the darkness and supplying light by continuing your specific implied actions which, by emitting light for others to see, brings hope.
(Candle is the original source of light for this meaning.)