Clean the lens
Turn off flash
Use the rear camera
Tap the moon to focus
Lower exposure by sliding down on the sun icon
Use 2x, 3x, or higher optical zoom if available
Avoid digital zoom as much as possible
Hold the phone steady
Use a tripod or rest the phone on a stable surface
Use a timer or shutter delay
Shoot during a clear night
Try when the moon is bright but not full
Turn on Night mode if needed
Take multiple shots
Use a manual camera app if available
Edit brightness and contrast after taking the photo