The moon DOES affect weather but not significantly. Just like it affects ocean tides, it affects the atmosphere in a similiar way.
When the moon is full or new for example, it creates a "bulge" in an ocean, which is why we have tides. This is due to the combined gravitational pull of the moon and the Sun.
There are also atmospheric tides, but lunar atmospheric tides are very weak. To detect a lunar signal in weather patterns can be difficult, because other signals are so much larger, and there is noise too. So if your friend thinks it's only been good weather during full moons, that's either a coincidence or, most likely, confirmation bias.