If an L1 space station was used as a staging area for a mission to Mars, it looks like we would use the Oberth effect in a close flyby to Earth to make the burn.
But I just realized - the idea, as stated, has a free degree of symmetry. I imagine that we need to leave (v_infinity) with a velocity vector in the direction of Earth's orbit (as per the Hohmann transfer orbit). We have to stay in the elliptic plane. That means there are 2 well-timed passes that could accomplish this in a month (neglecting the position of Mars). Standing on the moon's north pole, looking at the Earth, the trajectory could either fly on the left of right of Earth. Put another way, the flyby could be prograde or retrograde.
One trajectory would obviously be better than the other, but which one?
Obviously one factor would be the velocity of the moon itself. But maybe the gravitational potential relative to the sun would factor in as well? Initially thinking about it, the retrograde option would seem to start at a higher solar gravitational potential.