The SYHA Edinburgh Central is not bad. Usually fairly quiet, not too many actual yoofs. Option of private rooms if you want. Can feel like a bit of a hike up and down Leith Walk to get there, but plenty of buses passing by.
Or the SYHA have a summer hostel, in various student flats. So gives you a private room. Usually on the Cowgate, so very central. Though maybe noisy with drunks wandering past all night...
Or you can book a room in the student halls of residence. Not too far to walk from the centre.