The reason there is a difference in how much light is reflected, is because the moon is in a different position. During a new moon, it’s on the day side of the earth (so in between the sun and the earth) while during a full moon, it’s on the night side.
So there could theoretically be a gravitational effect, although I don’t think it would impact anyone’s sleep.
If anyone is noticing a difference in sleep quality, it’s most likely indeed to do with the amount of light.