IOS – Contact names are gone, but still shown in Messages

A custom label will have to suffice here. I have many contacts with their cell phones listed as 'mobile', and then a custom 'Google Voice' labeled number.

[edit]
An alternative to this is to make the change via Address Book.app on your computer, then sync the contacts to your phone.

Custom labels are available in all contact adding/modification screens.

First, browse to a contact, and tap the label name (to the left of a phone number, e-mail address, etc.). You will see the list of labels.

Second, swipe all the way to the bottom, and tap on "Add Custom Label".

Type in a new Custom Label;

And then it will be displayed next to the number.

It will now appear in a list of custom labels on later access.

Just to give post-iOS 7 input, the contact situation is still problematic. If a Facebook friend changes their last name to something else (or uses a joke name), it will cause sync problems with your contacts.

You can “Link” contact cards together (edit one contact, go to the bottom, hit Link Contacts, and connect it to another card). This usually keeps the contact info connected even if the name on Facebook changes, but in my own, personal experience, contact sync errors happen when it tries to update from Facebook when you have a poor cell/wifi connection. But that may be unrelated.

A side note: I found a bug in iOS 7 though; when you link personal cards (i.e. your information from Facebook to your personal card that Siri uses), deleting duplicate or outdated fields will cause the Contacts or Phone app to crash. So if you link your own contact cards together and have to make an edit, you have to unlink them first.