In the movie John Carter, what did the men discover in the mine?

The box was empty.

After years of searching, and probably so that he would eventually have someone to guard his body on earth, he used the fake discovery of the box in order to draw in his watcher who would want to take the "discovered" medallion. But the box was a trick to draw in the watcher and kill him and then take his medallion and return to Mars.

As indicated in the prologue and the epilogue, John Carter had spent years searching and digging for Thern artifacts; medallions, another "spider cave", whatever. As Carter also related, he was also aware the Thern agents had him under surveillance.

My takeaway of the discovery at the excavation was Carter finally making his move. Whatever pieces he required were in place (the fancy crypt, Ned being old enough, and who knows what else) and generally his patience finally running out. All he needed was one of his sites to yield up a bit of excitement. What the men found was entirely irrelevant, as long as they were properly exuberant. Carter took the thing, made it into a tantalizing top secret mystery to drive the Thern spies to a fever of curiosity, and he thus put his endgame into motion.

That's how I interpreted events, anyway.