This is another classic example of misinformation that has no basis in fact. The bottom line is this: you can absolutely see the northern lights during all phases of the lunar cycle. The primary factor here is that a full moon can make it more difficult to see weak auroras, but I am a firm believer that a full moon, combined with powerful auroras, which are often present during solar maximum is one of the finest combinations, a sky watcher could ever ask for.
The reason for this is simple: the bright moonlight illuminates the surroundings and allows you to see incredibly dimly lit landscapes with beautiful auroras dancing overhead. Our company offers lots of photographic expeditions with the sole focus of capturing pictures of the northern lights, and I can see from experience that a large percentage of professional photographers actually choose to travel with us during a full moon because it is such a photogenic and spectacular combination.