It might affect climate, but not on the time scale of a month, and does not significantly affect the weather.
The fact that the moon exists may significantly stabilise the inclination of the Earth relative to the Sun. This, in turn, affects climate in the long run. The debate is ongoing. For example, see long term axial tilt (Wikipedia):
The Moon has a stabilizing effect on Earth's obliquity. Frequency map analysis suggests that, in the absence of the Moon, the obliquity can change rapidly due to orbital resonances and chaotic behavior of the Solar System, reaching as high as 90° in as little as a few million years. However, more recent numerical simulations suggest that even in the absence of the Moon, Earth's obliquity could be considerably more stable; varying only by about 20-25°. The Moon's stabilizing effect will continue for less than 2 billion years. If the Moon continues to recede from the Earth due to tidal acceleration, resonances may occur which will cause large oscillations of the obliquity.
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 confirmation bias.