One thing to consider is a ghost tour.
While I do not believe in ghosts, there are certainly many strange and eerie places in Edinburgh, particulalry in the old town near the castle.
When the plague hit the narrow crowded lanes in Edinburgh the authorities decided to brick up either end of some of them and allow the people inside to die.
It is said ghosts walk these lanes.
As I said, I dont believe in them, but these tours do take you into some places that you cannot normally visit, and if you go late at night it can really scare you.
One of the most famous "haunted" areas is Mary King Close. More here
http://en.wikipedia.org/wiki/Mary_King's_Close
http://www.stuckonscotland.co.uk/edinburgh/mar y-kings-close.html
http://www.realmarykingsclose.com/
http://www.mercattours.com/home.asp