Ok, I tried everything on this forum to no avail. After messing with my phone last night I was finally able to find a fix that sticks. First, we need to know if your phone shows (Group) in the upper left hand corner of the contact app. If it does, then you will need to go to Settings, Mail/Contacts/Calendars, and click on each account listed, then disable contacts. It will ask if you want to delete them from your phone, confirm. (Only do this if you know they are backed-up). Afterwards, close the Contacts app and reopen. Ensure you now see a refresh arrow where (Groups) was in the upper left hand corner of the screen. If you do, go back to Mail/Contacts/Calendars under settings and re-enable the Contacts under each account you would like. Close the contacts app again and restart. Now, select (Groups) and uncheck the individual groups and click done. Then click on (Groups) again and re-check the individual groups, click done. You will now be able to search. Perform a soft restart by holding the top button and the menu button until you see the apple icon and you should be fixed.
If you already see the (Refresh Arrow) in the upper left hand corner of the Contacts App. Perform the same steps only instead of disabling the contacts under each account. Enable them, perform the steps after and end with disabling the contacts under the accounts.