Unutterably Immense Aurora Borealis
Credit: nasa.gov -- Aurorae on Jupiter
If you don't already know, aurorae are electromagnetic spectacles caused by high-energy, solar-charged particles bombarding with an atmosphere. They come in many shapes, magnitudes, and colors. Here's how we'll construct your shape-shifting moon with them:
Multicolored Aurorae
Because you've now allowed for an atmosphere, we can actually design how our aurorae will appear. Atmospheric composition determines the color of the aurorae as those high-energy ions interact with the various gases involved. Atomic oxygen produces red, orange, or green light depending on the amount of energy it becomes excited by (generally green light at lower altitudes, more-so ruddy light at higher altitudes in Earth's specific atmosphere). At much lower altitudes, atomic oxygen (again, specific to Earth's atmosphere) becomes less common (atomic oxygen is produced when diatomic oxygen drifts into the upper stratosphere and is broken by UV radiation); nitrogen instead can be used and excited: nitrogen absorbing energy will produce blue light, while nitrogen losing energy (after having gained excess) will produce red light.
Aurorae manifest in curtains descending into the atmosphere, where atmospheric composition, descending into the atmosphere, changes as well. This results in regions where the colors mix: pink, a mixture of red and green light; and yellow, a mixture of green and red (greater intensity followed by lesser intensity).
We have a lot more colors to work with, however (keep in mind, lighter gases are likely to escape the atmosphere, so their concentrations will naturally be diminished [one should also factor in relative abundance under planetary formation conditions]): Helium, white to orange; Neon, red-orange; Argon, violet to a lavender blue; Water Vapor, dimmer pink or magenta; and Carbon Dioxide, blue-white to pink. Personally, I would select our basic, Earth-analog atmospheric concoction of Nitrogen and Oxygen, yet, with a greater proportion of Carbon Dioxide--much greater for the following reason:
Aurorae magnitude in relation to gravity, magnetosphere strength, and atmospheric mass
The magnetosphere of Earth is kind of weak. It manages to nudge only a small fraction of solar-charged particles to collide with the atmosphere. A stronger field will affect particle trajectories more, meaning more collisions with the atmosphere and more lights. Most of the particles flying off the Sun (besides neutrinos, I think) are electrons, which are negative in charge. If geomagnetic north sits at the north pole, then the aurorae will be stronger in the northern hemisphere, although, after some degree of intensity, this distinction may no longer matter as particles bombard most of the atmospheric surface pretty uniformly, in the sense of whether any particular region of atmosphere is exposed naked and non-bombarded. 473
The south pole will attract the remaining positively-charged particles--protons and alpha particles--however, they are lesser in their abundances. Withal, the north pole would be "brighter," or more-so populated with aurorae than the south pole, however, ideally the aurorae will span the whole moon, just being brighter in those regions. Onward.
A thicker, more massive atmosphere in tandem with a weaker gravitational pull is ideal for maximal aurorae activity. The weaker gravity will extend the atmosphere further from the moon's surface. The volumes of atmosphere where the solar-charged particles may interact will be extended under weaker gravity (the atmosphere will be taller).
To complete the picture
Our atmosphere may allow an observer to see the surface. Ideally, we want our aurorae to be consistent and it turns out that solar activity itself is pretty consistent and unvarying. If the surface is ever visible through the swathes of folding, shaping, shifting, entombing, iridescent ribbons, then we must make it indistinct. A volcanic past could achieve this.
Credit: earthscienceeducation.org -- Basalt Rock
A surface largely of basaltic and obsidian-like rock could probably accomplish this.
Non-repeating patterns
Now, aurorae typically follow magnetic field lines of the magnetosphere. Magnetosphere structures vary depending on a multitude of factors, such as the distance from the sun, the intensity of solar winds, among others. Magnetic geometry is not quite that simple, however. The Aurorae we should expect shouldn't fall into predictable routines, though, just to be safe, let's just offset it from its rotational axis and center of mass and make it do other weird stuff. Also, let's assume that our inhabited world also has a magnetosphere (a safe assumption?) of different, lesser strength and perhaps offset to another angle which isn't perpendicular to the equatorial (also a safe assumption that the moon orbits near the equatorial?). 188