As Candice says, Haymarket is rubbish if you want to do anything at all. But its on the way to Waverley, and that's probably where you'll end up changing train anyway; walk out of there and you'll find yourself either on Princes Street or in Princes Gardens, both of which are good for a gander. Head south from Princes Street and you'll hit the Royal Mile, which is by far the greater of the two streets to see. The Gardens are in between both.
You can't miss the Mile, its the street with the castle at the top and the Scottish Parliament (the one that looks like a post-apocalyptic bomb site) at the bottom. It contains most of the attractions to be seen in Edinburgh, right next to the station, many of which can be done in much less than an hour.
These include, and aren't limited to: The Camera Obscura, The Childrens' Museum, Edinburgh Castle, a very fascinating and modern science museum whose name eludes me, and many, many pubs...
The parliament is a bit far down to do in an hour, though if you have some extra time its right next to my favourite part of Edinburgh; Arthur's Seat.
Basically, find a map, find the castle, and everything from there all the way down to the bottom of the Royal Mile is fair game.
And just think; I know all this wonderful stuff, and yet I still hate the dang place. Get your changeover at Glasgow next time, we'll go for a pint

Edit: because I should know how to spell our own train stations' names properly...