Usually this problem may occur when contacts have been imported from a csv file that was exported from non apple machine or application. below solution might help you if you are luck. give it a try. I tried almost everything else before going for this solution.
here are steps:
1. Take a backup of all your contacts (either csv or gmail csv format).
2. If you already have a vcard format file for your contacts then skip to step 7 Otherwise, make a copy of file created in step 1.
3. Open the csv and try to remove all the spaces (if any) in the contacts field/column. (select column(s) and search and replace all for space character. make sure you do not remove space from column headers)
4. Save the csv and make sure you save it in csv format {click yes on first save dialog and no on 2nd dialog (if appears)}
5. Import this csv to outlook or address book.
6. Now Export the contacts imported in step 5 to vcard format.
7. Open the vcard file in a text editor and now you have to perform the bulk search and replace function.
8. You will search for TEL:0 and replace it with TEL:+<CountryCode>
9. Replace All
10. Save the vcard file.
11. You will need an icloud account now. Create if you dont have using your apple device.
12. If you already have icloud account then sign-in to it.
13. Click on Contacts
14. An book format page will appear.
15. You may create backup of your contacts which already exist in you icloud account (vcard format).
16. Now Click on any contact and press Select All (using keyboard).
17. You may either delete all these contacts OR create a new group for these contacts (if these are working fine).
18. Now click on settings icon again and click import (vcard).
19. Select the file you saved in step 10. Your contacts will be imported to iCloud.
20. Now the device part where you need to bring your contacts in correct format.
21. Please unlock you iPhone and goto settings, click on General, then click on International and see if your country is selected correctly in Region Format.
22. For iPhone-5 there is a setting in Settings > Phone where the phone assists for correct country code. Turn it on.
23. Now Open Settings and then Mail, Contacts, Calendars area,
24. Click on iCloud account (or add it if its not there, the account where you imported your contacts in step 19).
25. Turn on Contacts Sync.
26. Turn off the contacts of unwanted mail accounts (e.g. Exchange, Gmail, hotmail, yahoo etc) , since you now have all ur contacts in iCloud account.
27. Go back to home screen and click on mail icon and sync once.
28. Now Check if your contacts show name or not.
