One morning I overslept and discovered that it was because my alarm—my iPhones 3S, 4 and 5s have been my alarm clocks for years—didn't go off. And it didn't go off because despite having been plugged in all night, it was dead as a brick.
So, I, too, have this issue, and I have put a flashlight on the port on the phone and it appears to be clean and clear. But because I can plug my iPad Air into the very same cable and it charges just fine, I suspect it's something more than the cable.
I tried the workaround in which you push the plugged-in cable slightly to the left and affix it that way with a rubber band it to the phone, and it's worked, but it doesn't give me much hope that the cable will last that long or that the problem is in any way fixed. I bought the 5s when it came out, and despite thinking a few weeks ago that I had a battery problem (user error; stupid user!), I still have pretty good battery life—way better, still, than I ever had with the 4.
I have several lightning cables, having got one each with phone and iPad and having bought a couple extra for charging in the car, etc. Of those, I can charge the iPad on all, but can charge the phone normally on only one. This leads me to suspect that because I charge the phone more frequently than the iPad, there's something amiss with the port on the iPhone. Could it be an issue with the inside of the port being dirty in a way that I cannot see? Some sort of glaze built up from repeated plugging and unplugging?
Anyone have any thoughts or done any experiments along those lines?