In my view it would be Easier to change the moon then Mars. All the moon needs is a strong magnetic field and it would build its own atmosphere. Dust crashes into the moon all the time and some of the dust contains gases. The solar wind knocks the gass molecules away from the moon's surface because it doesn't have a magnetic field. The moon Titan is only slightly bigger than Earth's Moon but it has a thick atmosphere because it is inside Saturn's magnetic field and it is far from the Sun the solar wind is weaker there.
Imagined that if the moon had a weak magnetic field it may link up with the Earth's magnetic field and form a bigger magnetic field and that would just be enough to deflect the solar wind particles. Then as dust and Gases collide with the moon they would stay there and build up the moon's atmosphere. Unfortunately the moon is outside Earth's magnetic field right now so it has no shield.
