What would cause the ceiling fan light to blink or shut off?

Filament a likely culprit, but is the fan perfectly balanced? We all assume not perfectly. Bad connections, even good ones, break with enough movement over time. Potential chain of events:

Power on - rush of power bridges air gap, or gap doesn't exist, connection holds together

Running - something gets hot or vibrates loose, loses connection

Stop - cooling or lack of vibration allows physical gap to close enough so that...

Power on - rush of power bridges air gap, connection holds together, repeat

The connection could be filament to post, switch to wiring, internal in the fan, the place where a rat ate away connection material, anywhere in the current loop really, but filament, switch, and internal to fan are my top three suspects.

On old fixtures I have found the center contact to be the problem many times. With the power off remove the lamp, use a small screwdriver to lift the center contact about 1/8" replace lamp and see if that fixes it. If lifting the center contact fails next I would be looking at the wiring on the fan, the switches used in fans even old ones are usually quite cheap and can fail as a loose connection also. I would check the socket first as this is quick & easy and has solved a large number of light fixture problems for me in the past.