The threads in the door knob have worn out (stripped threads) and the knob came off. I am not sure how to fix it back other than supergluing it back on. What is the right way to fix it? I am thinking that replacing the screw with a thicker one might not be the way.
Put some wood glue inside the hole, put 3-4 toothpicks (typically made out of bamboo, very strong wood), wait until dry. Screw knob back on.
The toothpick will provide both the filler and the strength to allow the knob to stay on. The screw will have no problems gripping on the bamboo reinforced hole, and the bamboo will be glued to the wood by the wood glue and also the bamboo digging into the wood. This method is a fairly strong repair.
The permanent fix would be to put a threaded metal insert into that knob, sized to math the threads of the existing bolt (or to get a matching bolt/insert pair). Wooden threads holding a knob that people are tempted to rotate will always wear out sooner or later.
Or glue in an appropriate sized hardwood dowel to plug the hole and drill a new hole that isn't worn out.
Or try the trick @solarmike mentioned. I would recommend toothpicks rather than wooden matchsticks; they tend to be harder wood and hence more durable.