I just had a slightly confusing experience changing the iCloud account on an iPad mini. I was helping out a family member who had recently changed their Apple ID email address recently, due to a change in ISP/email provider. In the iPad settings the ID showing in the “iTunes and App Store” section was correct, but the ID showing in the iCloud settings was the old ID. This wasn’t a problem until recently when the iPad started prompting for the password for the iCloud account every 30 seconds, rendering the device unusable.
It wasn’t at all clear how to change the iCloud ID. Clicking on the ID at the top of the page only gave the option of entering the password for the old incorrect ID, with no ability to change the ID.
Eventually I discovered that I needed to scroll the page up to reveal the “Sign Out” button at the bottom of the page. After signing out, I was then able to sign-in with the new correct Apple ID. Simple once you know what’s going on.