AirPods not charging due to contact issue between the AirPod and the case
Try these solutions in the following order:


Cleaning the lightning port*;
Cleaning the inside of the case*;
Instead of manually inserting the AirPods in the charging case all the way, insert just a portion of them, and let them fall in the case. The gravity and the magnetic attraction will place them in the right position, thus allowing them to charge. Surprisingly, inserting them manually all the way might not place them in the right position;
Place a small piece of paper on the inside of lid so that when it is closed, it presses the AirPods against the connector**;


If none of these options worked, the case and/or the AirPods might be broken and need to be looked at by an authorized technician




*with a sharp object that doesn't conduct electricity (like a toothpick)
** This solution should be used as a last resort, since it might potentially damage the opening/closing mechanics of the lid