Personally I'd avoid metal tools for cleaning it out as if you're not careful you can damage the socket. I just cut a thin slither off a bamboo skewer and use that - you can get a pack of 100 skewers for next to nothing at the supermarket and that'll last you forever - unless you actually make kebabs with them :D

I use a toothpick to clean the ligthning port of my iPhones. Toothpicks are cheap as well..

This. So many times this has been the cause of the problem. First time someone pointed this out I was amazed at the amount of lint I pulled out and even more amazed that the port was - with some jiggling - able to function at all with all that crap in it.

Huh. I keep my phone in my pants pocket, and I never once thought about lint getting in that socket.