I have stayed at the Alpek Hotel in the Eminönü part of Istanbul multiple times. What you get here is a clean, comfortable, but fairly small room in a spectacular location within steps of fantastic shopping, a zillion restaurants, ferries across the Bosporus and a kind of friendly relaxed charm that is hard to describe. If you need a huge room, go somewhere else, but otherwise, enjoy the excellent breakfast on the roof as you begin the day watching ferries come and go in the Golden horn, marvel at the Yeni Cami Mosque next door and watch the world go past on a parade of ships through the Bosporus. This is a truly special place.
The major tourist attractions are mostly within walking distance, although the walk can involve a lot of uphill. If you don't want to walk, the tram is right there to take you to Hagia Sophia, the Topkapi Palace, the Blue Mosque and the outstanding Archaeological museum.
I've stayed in many hotels in Istanbul, Alpek Hotel is far and away my favorite. It is small with old world charm, but a functioning elevator and staff who are wonderful. Don't necessarily expect this every time, but if they can they will let you into your room early and simply have a way of making you feel welcome. I'll be staying there again.