The Copper Mines trail up at the Water Gap is fantastic. There are a few old copper mines you can sort of explore; some of them have been gated so people don't get trapped. Most of the lower trail is built along a cart path used to haul the ore out of the mines.
The second upper mine you come to on the red blazed trail is the one I call "Proposal Rock." Apparently some spelunking romantic brought his girlfriend into the mine one day to read a plaque that says "will you marry me" on it. I wonder if she said yes...
The old Iron Mines in Hibernia are pretty neat also. The last time I was there, none of the mine entrances were officially gated, but there had been a few minor cave ins where rocks blocked the deeper recesses of the mines, so you could not go very far.
NJ has quite a history of mining; I was not aware of it until I read a guidebook about the Water Gap that discussed the history of 16th century Dutch miners in the area, that lent their distinctive names to so many of the places in the region.
One of the more interesting mines in NJ is not what one would traditionally consider a mine at all; it is a bog-iron ore pit in the Pinelands, near Batsto State Forest. Bog Iron was literally leached from mud dug out of the tea-water rivers in the pinelands. It created a bustling industry in the 1700s and was responsible for a portion of the settlement in that region. There is a great living history museum in Batsto devoted to the lives of the 18th century settlers of the village. Go see the old fashioned saw mill when it is in operation for a real treat.
Most of the mines in the area are iron mines. There are a few near & in Mahlon Dickerson. I would not recommend exploring them. I believe there is a mine in Kittatinny not too far from Route 206. Kind of cool.
And then there was a uranium mine in Allamuchy! Pass the yellow cake, please!