Some people were having trouble with an earth observation satellite. This particular one had a clever scanning system that used the satellite's own motion for one axis, and a pivoting mirror for the other. The problem was, as time went by, the mirror wasn't able to swing as far, so the swathe the satellite could observe would slowly get narrower and narrower.
Nobody really understood why this was happening, or how to deal with it. Eventually, they asked my dad to look at the problem. His approach was to get copies of all the engineering drawings and then ponder them, which he did for a couple of weeks. He eventually formed a logical hypothesis of why the mirror might be getting restricted. The mechanism that moved the mirror involved a collar that slide along a greased rod. My dad's idea was that the grease was getting thick after being in the vacuum of space for a while, and piling up on the ends of the rod, where it would get in the way. Better yet, he had an idea to fix this.
The procedure for putting the satellite into idle mode while it passed around the night side of the earth was to center the mirror. My dad's idea was to modify this procedure and instead rotate the mirror as far as it would go to one side and let it sit there overnight. This, he figured, would tend to slowly squish the grease out of the way. The next night, park it at the other side, alternately pushing the grease away at both ends.
This made sense to the people in charge, seemed harmless, and might work, so they approved the change in procedure. And sure enough, as the weeks went by, the mirror slowly increased its range of motion, and ultimately was capable of its full original scan.
So, from thousands of miles away, without even seeing or touching the satellite, my dad was able to repair it (and avoid the same problem in future satellites), by making a simple change in a procedure. All from looking at the design documents and thinking about it. Good on ya, dad.