Botanic Gardens if you are into gardens. Their restaurant also does a very nice afternoon tea.
The National Trust for Scotland Georgian house on Charlotte Square, which shows you how the well to do lived when the New Town was built.
Dynamic Earth is done very well -
http://www.dynamicearth.co.uk
I don't think anyone has mentioned Holyrood yet either - it can easily be combined with Dynamic Earth and the Parliament buildings (although the Parliament might be of more interest to Scots, given we fought so hard to get it).
And just be aware of the whole Independence debate Scotland is having at the moment - culminating in a referendum in September. You too can join in the national game of spotting bias in the media
