Basically that is an eye pimple ... and some do in fact last for a long period of time. I had one last year inside my upper lid.. I used a warm compress several times a time and eventually as gross as it sounds, the "yuck" in it oozed out..... It lasted for a couple of weeks..
the hot compress really made it feel better and did get it to drain.. I am surprised if the doctor didn't tell you to do that... BTW, not true what the Pediatrician told you............ it was my eye doctor who told me about the hot compress.. Also, IF you want.. the eye doc will give you meds for it..
Definitely take your son to the eye doc... those Chalazion are SUPER common..