Thank you for the question. The solution to this problem that has been proven to always work is to perform an iCloud
backup of the phone, restore the phone to factory settings, and then restore the iCloud backup.
Some folks have been successful however by doing the following:
1) Double-tap the home button on your device two times, or until a menu of apps appears on the bottom of your screen. The home button is the physical round button on the front of the device with the white square inside it.
2) Once the menu of apps slides up from the bottom of the screen, hold your finger down on one of the apps. This will make a red minus (-) sign appear over the top of each application.
3) Tap the minus sign over the phone app to close the app.
4) Press the Home button again. This should return you back to your home screen.
5) Turn the phone off and wait 2 minutes
6) Turn the phone back on and test the caller ID again
Otherwise you do need to use the reset/restore method to fix the problem. I understand that is not the ideal solution, but it is the solution nonetheless. I hope the other method works, otherwise just make sure you take an iCloud backup before doing the reset. Please let me know if you have any questions and I will be happy to assist further. Thanks again!