I have an iPhone 6 and bought AppleCare for it. The Lightning cable that came with it is starting to split (see the picture). It still charges fine, to the best of my knowledge, but is it eligible for a free replacement under AppleCare?
I have had the same situation happen to me with 2 phones actually. Apple did replace them under warranty.
If you have access to an Apple Store:
They will then provide you with a new cable after verifying your eligibility.
If you do not have an Apple Store that you can get to, call AppleCare for instructions.
I have exchanged at least 7 of them in the timespan of 3 years dating back since July 2014, ranging from iPhone 5 to iPhone 6. And in my experience they do it for cables that do not charge anymore since they test that sometimes, but there are times that the person doesn't.
One time they told me that when rips happen, it's not a defect by the cable but more of a misuse by the customer but he still let me replace it since it was the first replacement for my iPhone on file.
They do look for the IMEI from your Apple device to see if it's still under warranty and not the cable's serial number.
You should set up an appointment for the genius bar since sometimes the wait is long.
Update for July 4 2017:
I recently went in with my iPhone 7 to replace a lightning cable. I was denied because the cable I brought in had a serial number that did not belong to the iPhone 7. It seems like they are more stringent on the replacements and you have to start tagging which lightning cables belong to which iPhone. My story is similar to this one: https://www.reddit.com/r/apple/comments/3t47uh/lightning_cable_replacement_policy_changed
This is a design flaw of the cable. The phone end is with a sharp profile change resulting in a stress concentration leading to the premature failure. It seems to be an intentional one so that one keeps buying the overpriced cable multiple number of times. I have several Apple devices, all of the cables that came with them have invariably failed. It has nothing to do with the handling as someone in here has said.
if it is a hardware fail they will replace it but if you broke it then they wont do it for free.