Here' s one of the advantages of having a 2-year-old and 17-year-old at the same time. When the 2-year-old gets your Chapstick and turns the bottom dial until the lip balm squishes into the lid, your 17 year old has the spare brain cells to come up with a solution. (Of course, we won't discuss why I don't have the spare brain cells.)
Since I had a couple sticks of lip balm that my young one had screwed up and the lids were packed with the stuff, I thought I would try my teenager's solution.
Put the lid in the microwave and melt the lip balm. I tried this with Chapstick brand lip balm, and another lip balm I got as a sample in a whole foods type store. They melted at very different times--the Chapstick took much longer. Just start with 20 seconds or so and see how it goes.
Screw the lip balm down a little bit, and pour the liquid back in.
You may want to pop it in the fridge to speed up the solidification. You may also want to aid the shaping a little with a finger. It doesn't save you a lot of lip balm, but it makes you feel better each time you take off the lid and no longer have the lid stuck full of the stuff.