Apparently, this is a problem many have already encountered and solved. I have not stumbled across it so far. Here is what my problem was:

At some point in time of using my iPhone 6 (a bit more than half a year old), didn't charge correctly anymore. It happened that in the morning, after supposedly charging it the whole night, the iPhone was almost empty. I then realised that the plug did not go fully in the port. Moreover, it only charged when the plug was slightly skewed in the port.

My initial thought was that this must be a defect with the Lightning port in my iPhone. Therefore, I went to my local GRAVIS store in Stuttgart (which is in walking distance from my home). And the employee there said that he could clean it for 10€. I was a bit surprised that he fully ruled out a port defect. He answered that in 95 % of times, this is from dirt in the port.

I tried his hint to use a toothpick to clean the port. I found crazy amounts of dust in the port. And now it is fully functional again. Sometimes, problems are much easier to solve than expected.

A quick Web search showed that I was definitely not the first with this problem. For example, Dan Graziano on CNET also describes the toothpick hint:

Lint and other debris from our pockets or purses gets stuck in this port and can build up over time. To be safe, back up your iPhone or iPad to your computer or the cloud. Power it off, and with a normal toothpick, gently remove the lint. You will be amazed how much can get stuck in the port.

As I hadn't seen this solution before, maybe this article is helpful to somebody.