The “Linked contacts” feature in iOS 6 was kind of fuzzy and depended on Facebook's integration. Things have changed in iOS 7 so you can merge contacts any way you like easily.
Apple has made sure that the “linked contacts” option is a default editable option for all contacts on your iPhone/iPad/iPod Touch. You can use this to link/merge contacts and the result is just the same: only, simpler and faster.
How to Merge Contacts in iOS 7 Using the “Linked Contacts” Feature
Step #1. Open the Phone app and tap on Contacts (on the iPad, tap on Contacts app.)
Step #2. Now, tap on a contact that you want to merge with some other contact.
Step #3. Tap Edit from the top-right.
Step #4. Scroll down and you should see “Linked Contacts.”
Step #5. Tap on “link contacts…“
Step #6. Choose the other contact you want to merge.
Step #7. Tap on Link.
Step #8. Once the contact gets added, tap on Done to finish editing the contact
You should note here that when you merge contacts this way, the contact that got merged will not show up in the contacts list. It just gets merged – but strangely, if the contact is merged and synced in iCloud, it will be a separate entry in iCloud (as far as our testing goes).
As a result of this, the merging does not get reflected on other devices that you use even when you've synced contacts with iCloud.
To change the name you can either simply edit or tap on the linked contact (from under Linked Contacts) and tap on “Use This Name for Unified Card”
Warning: the only thing you should remember is that the merging is device-specific only. No iCloud sync happens with the merge so you have to be careful not to delete any duplicate contact that has been linked. If you want to truly “merge”, you have to copy the details of the duplicate contact into the original one.