Right where the action is!
I know of a hotel in a very good location in the historic quarter of Sultanahmet, 5 minutes walk from the Blue Mosque:
http://www.sultansinn.com/
And their sister hotel right opposite (off the Sea of Marmara!):
http://www.denizkonakhotel.com/rates.asp
The 2nd one offers great views, service, clean rooms, modern & large bathrooms, good breakfast, all for a very modest cost!
I visited the 2nd one as a couple I befriended was staying there; then I too moved there from a pricey hotel! Rooftop restaurant where you can sit in the covered areas or open-air, admiring the Blue Mosque & surroundings.
Complimentary slippers, and decent first-aid kit!
Lovely room balconies with views of the sea & the gardens in front of it!
