From what I know, they have an extremely powerful influence over magick done that night. Magickal workings (mainlly dealing with the goddess and the things the goddess watches over) under a blue moon will have alot more energy than usual.
Divination, prophecys, and generally preditions of the future are one of the major thing influenced by the blue moon, especially if done outside or used with lunar corrosponding things. They often, so I heard and seen for myself, are much clearer and more acurate. The blue moon's energy is generally used in spells for protection, love, wisdom, and from what Ive heard, communication with spirits.
(I'm not really sure if this is true or not, but I once read somewhere that a blue moon's appearing cracks open the astral realm to the mundian world just a little bit wider temporarilly. That book also said that because the astral at the time is more accecible, it is much easier to contact astral spirits and elementals, namely sylphs and fairies. Because of that some even nick-named the blue moon "the fairy moon." But still, Im not really sure if this is true or not.)