The moon, combined with a special body of water or magical object, often causes the initial mermaid transformation, as in H2O.
Along with moonstriking, the full moon throws Holly and Claire's powers out of whack (such as Holly demonstrating cryokinesis or Claire pyrokinesis without having these powers before) as aftereffects a few days later.
For Hailey, Brooklyn and Merlena, moonstriking was rather ambiguous: the obvious reaction was unnaturally white faces for the duration of the night, but the morning after, the girls used their powers for the first time. Whether this was coincidental or caused by the moon is unclear.
Moonlight tends to have a strange effect on mermaids. They become "moonstruck," acting wild, seductive, flippant or otherwise out-of-character, and sometimes gain unusual powers. Again, this primarily comes from H2O (and thus the full moon), though there is a mythological link between the moon and temporary insanity.