This seems to be more than a faulty cable issue. I am in the cell phone repairs business and I have received many calls in the past couple of weeks about iPhone 5 no longer charging once the battery has completely drained.
At first, I thought that it was a faulty lightning connector and I replaced it only to find out that the phone would still not charge. Next I replaced the battery and the phone powered up (battery was pre-charged) and started charging normally. However, after a few days, the battery was allowed to completely drained again and the phone would not charge again. I doubted that it was a faulty battery but I replaced it again just in case it happened to be defective. Same thing happened with the second new battery, the phone worked and charged properly until the battery was allowed to drain and then the phone would not charge.
The only solution that I have so far is to take the battery out and manually charging it using an 5 volts external power supply. I only have to mannually charge it for a few minutes, enough to get 2-3%. After a few minutes, I put the battery back in the phone and the phone powers and charges fine.
I know that this solution is not practical for most but it works. To me it appears that this might be a software or a power management issue with the motherboard.