I think the iMessage App mismatches the phone number notation with the Contacts App on iPad. Maybe it's a bug in iOS 7.x.x, I don't know.
OK, I think I found a solution:
My language of the iPad is different from the language I speak. Hence the notation of phone numbers is also different.
Your Apple ID is used in a certain country with its own notation of phone numbers: Settings>Messages>Send&Receive>Apple ID. Tap your Apple ID and then tap Change Location to find out which locattion iMessage set to.
Then go to Settings>General>International>Language. Set the language according to your language iMessage is set to. Also, change the Region Format to the location of iMessage.
Restart the iPad and for my all my contact names appeared, instead of their phone numbers. Now all is set and you can change your language and region format back, still keeping your contact names in iMessage.
Hope this helps.