The old graphite mines at Seathwaite in Borrowdale are interesting to visit, even more so if combined with a visit to the Pencil Factory Museum in Keswick (on a wet and windy day).
There is lots of info about the Seathwaite graphite mines on the Internet. Just Google: seathwaite plumbago mine
Some good photos of the mines here:
http://www.markrichards.info/publications/lakeland-fellranger/the-western-fells/grey-knotts_1.htmlA good walk to do to take them in would be to start at the National Trust car park at Seatoller and walk along the road towards Seathwaite. At Seathwaite Bridge take the path on your right and follow it past the ancient Borrowdale Yews (read up about them first) to reach Newhouse Gill. Here you'll find a track, which isn't marked on the map, which heads uphill towards the various mine entrances. You can then continue to climb uphill towards Grey Knotts and then follow the fence line to Brandreth. From there you can head down towards Dubs Bottom and Little Round How (maybe include a quick out and back to Haystacks) and then head east along the old Drum track to Honister Slate Mine (have a look at the history display in the shop at the mine). From Honister, head downhill in the direction of Seatoller, following the road first before picking up the old mine road to Seatoller.
Not sure of the distance, but it's a walk I've done lots of times and can take a full day by the time you've looked at everything.