Earthshine depends upon the Earths albedo (reflectivity).
At the time around the New Moon, (from the perspective of the surface of the moon that faces the Earth) the Earth will be lit at a maximum – thus will reflect back to the moons surface at a maximum.
Spring for the Northern hemisphere has on average the highest albedo of the year.
Snow and Ice has the most affect upon albedo (far more so than water).
The affect is also seen at a maximum at moonrise and set.