If there's dust/hair/lint in the connector (and there almost certainly is; I've seen Apple Geniuses fish an enormous amount of junk out of those connectors. Sticking a Lightning plug into the socket just packs the gunk down into the socket until you can't charge at all), then it's much safer to get it out with a plastic tool, which won't carry a static charge. Like this one:
I wish Apple would come clean about this issue, because this is a problem that can occur with normal use. If you stick your phone in a pocket or purse, small particles and lint will go into whatever opening the phone presents. Isn't that how everybody carries their phone?
Unfortunately, Apple's support note has a 'refer to qualified service personnel' disclaimer:
"If your battery doesn't charge, or if your device shows a low-battery image, black screen, or "accessory not supported" alert, follow these steps.
Inspect the connector at the bottom of the phone:
Make sure that there are no bent pins in the connector.
Make sure that the connector is free from debris.
If there are bent pins or debris, please contact Apple Support.
Otherwise, continue troubleshooting."