After updating my Apple Watch to watchOS 3 some of my contact names are missing in Messages and Phone etc. Some are showing name, others just show the phone number. The iPhone does show the contact names where the watch doesn't.
This answer's a bit late, but I finally figured out the issue.
Your Apple Watch's contact list is most likely out of sync with your iPhone.
Try checking your Apple Watch's contact list though the phone app. If it looks like some contacts are missing, it's likely that the contact list isn't synced properly.
The easiest way to manually re-sync your contact list is to create a dummy contact "AAAAA" on your iPhone. Once it's been resynced, all your contacts should start displaying properly again.