When it's rainy, you can only find yellow crabs (Parcheeks) and Momotis. In order to make it stop raining, you MUST sacrifice one (or more) of your patapons. Notice the cap in the top right corner.
Keep refreshing the weather of that stage (keep joining then retiring from any mission), until its sunny. To make sure that you have done this correctly, enter the stage, and there should only be a golden bird (Motimotitsi) and a creature with wings and a oversized beak (Picheek).
Kill the Picheek, (at the start it looks like a Kacheek ) and a star called Hoshipon should come out.
Keep Hoshipon in the middle of the screen, don't move too far ahead and collect the Ka-ching, or it will fly away. Drum any beat once if Hoshipon asks if you are listening to him.
After he finishes his speech, he will drop a Black Star.
After that, repeat step 1-3 until its sunny again (this may take a while).
Kill the strange looking creature again.
Keep Hoshipon in the middle of the screen, don't move too far ahead.