Finally, finally! I've got the real answer! After "fighting" with my iPhone 5s for the better part of 6 months, I've got the answer to why my iPhone always reboots overnight while charging, and happens several times during the day, if it's plugged in. I have tried software fixes, resetting everything. I've had a new battery installed. All this, to no avail.
You ready for this? it was some foreign particulate matter in the charging (lighting) port, and not lint. I had taken it to the Apple Genius and they said they cleaned out the port although they said it wasn't all that dirty. It still would do these random reboots. I then took it to a small Mom and Pop computer store that specializes in Macs and iPhones....in a second, he said, there's something like graphite in the port; he felt the irregularity with some sharp tweezers; he said, it wasn't "flush". I thought he'd damage the port, but he assured me these ports are heartier than one thinks. He removed (scraped out) the little piece caught in the back of the port, and lo! and behold! No more reboots. So, yah have to find someone who really knows the iPhone innards. Ahhhh, life is good again. :-)