exspello's suggestions are excellent. However if you are prepared to venture further into the Peak District there are the more commercial options of Castleton, namely, Treak Cliff, Blue John, Peak Cavern and Speedwell Mine which has the added attraction of part of the trip in a boat.
Buxton too should not be overlooked. Poole's Cavern is also well worth a trip.
There is also the option if you think it's for you and your son of joining a group or society such as the Peak District Mines Historical Society. PDMHS gives, through its newsletter and website, members the chance to join both underground and overground trips to non commercial venues. If you go to Temple Mine and the Museum you will be able to get full details there.
While you are in the Peak District there are a couple of freebie above ground sites that are well worth a visit. Magpie Mine near Sheldon and High Rake Mine at Windmill, which is on the road between Tideswell and Bradwell
Tread carefully and take care, Peter.