I got that that it works ok in normal conditions and the problem is extra fan cycles when they should not occur. It could be the t-stat giving a fan-only signal, but regular and repeated fan on/off at 60 second intervals sounds a bit beyond what failing thermostat might do.
The a/c control board has all the possibilities of the thermostat (afterall, it is what the thermostat talks to), plus some more of its own. In addition to doing fan-only mode, it can cycle the fan without running the compressor if the internal compressor freeze-up sensor trips, and it also does it as a part of normal time delay after any heat or cool cycle.