Spending four nights and five days in Istanbul in October of 2012 was about perfect to get a feel for the city. We stayed at the Hotel Ibraham Pasha located right off of the Hippodrome and with a fantastic roof top bar (hence the large number of pictures of the Blue Mosque at various times).
Sights included some things we didn’t expect, including hair and teeth fragments of the Prophet, and the Staff of Moses.
If you’re there for more than couple of days, do get the Museum Pass - while it may not save you money, it certainly saves time as you don’t have to wait in lines for entry tickets.