Staying in a good but reasonably priced hotel was very important to me on my recent trip to Istanbul. I did a good amount of research, and it paid off, as I chose the Armada Hotel and couldn't be happier with the result. The room was clean and comfortable; the queen-sized bed was firm and had 4 fluffy pillows! Breakfast at the rooftop restaurant was spectacular; the food was great, the view top-notch, and the hotel made sure I had an English language newspaper waiting for me at breakfast.
I went in the off-season, but I was pleased that the hotel is tucked away in a quieter section down the hill from Sultanahmet. It's also right across the street from the Bosporus, which made for relaxing evening walks out to the fish restaurants at Kumkapi. It is a rather steep uphill climb back to Sultanahmet, though, but taken slow, the walk is pleasant enough.
That whole area behind the Blue Mosque and down to the Bosporus contains some great restaurants and good hotel bars. The staff at the Armada is very helpful and the whole place exudes a nice sense of calm and comfort that's quite welcome after a busy day in a boisterous city.
The bathrooms were spotless (another important consideration for me), the TV worked (but was a bit small) and the fridge comes stocked with food and drink. The rooms contain copies of Cornucopia, a great magazine, as well as other reading material. I would certainly stay again at the Armada.