There is also the Side Hotel in Sultanahmet. The stuff can be a little dodgy, esp. if you are a girl (I had to throw the receptionist off my bed and out of my room at 2am after an overseas flight because he insisted on playing his i-tunes to me and offered to massage my feet) after an overseas flight. >; P BUT it is cheap, yeah. Like 20 dollars a night or so? They have a website and you can book online. Nice view from the breakfast room on the roof, and just a few minutes from Agya Sofia , Four season hotel and the Blue Mosque.
You might also be able to get a room at soundfabrika, an Oriental music school for foreigners near Tunel. Simple, but at least there are other foregners. They too have a website.
You will have less hastle if you stay near Taksim as that's where the Turks go, less tourists, no carpet touts at all. So Cemberlitas is a good bet too, as Tania said.
What are you planning to do in Istanbul?
If it is your first time in Turkey, make it a basic assumption that people will try to rip you off, overcharge you, sell you stuff you don't need and the constant offers for massaging your shoulders and other parts of your body...be careful with your money. As I said, Taksim and Istiklal area can be less hastle and feel more normal than Sultanahmet where all the tourist attractions are. Plus most of the night life is here on Istiklal.