I stayed twice in Sokullu Pasha Suite, a top floor apartment in an old and authentic neighborhood of Sultanahmet. Nothing glamorous in this apartment: two bedrooms, one for parents with a double bed, another for kids with two single beds. The kids' bedroom has an access to a large terrace. There is a living room with a kitchenette too. Furniture and kitchenware are scarse, but they will do if you stay for not more than a couple of nights. The neighborhood is quiet and safe. All the pearls of Sultanahmet, such as Topkapi Palace, Aya Sofya Mosque/Church, Egyptian Column, Binbirdirek Cistern, are within walking distance.
I wouldn't recommend it for elderly people because there is no elevator there. I do recommend it for a family with kids, who are on a tight budget.
One more advice - do ask for transfer from the airport, or else you may not find the place easily in the labyrinth of alleys.
I shall certainly stay there again when I go to Istanbul with my family or in a company of friends

There are actually two buildings, the main one where you eat breakfast, which is further down the hill towards the water, and the one further up the hill next to Sokollu Mehmet Pasha Mosque, which inside has more impressive blue tile work than the much larger Blue Mosque. We stayed in the two bedroom top floor apt. Pros: large master bedroom with sea views, decent-sized balcony, full kitchen, good heating and hot water, clean, and location (bottom of Kucuk ayasofya mahallesi su terazisi, so all you need to do is walk up the steep hill a few blocks and turn right, and you are in the middle of Sultanahmet!) Cons: If u walk from metro station, you will never find the main building where you check in - we were lucky to find a man named Ismael (salt & pepper hair, mid 50's) who hangs out near the Sultanamet metro asking if u need a hotel or help finding yours - for 10 TL it was well-worth it - 20 minutes from station in winding streets that look poor and scary - but infact are very safe. Biggest issue was call to prayer is deafening and will wake you up at 515 am, neighborhood is rundown but improving, especially up su terazisi street (have a sunset drink at Hotel Dara, 1 of the many new boutique hotels 2 minute walk away), you will need to walk 5 minutes to breakfast, which when the weather is good is ok, but in the cold rain, YUCK! The fridge is dorm-size and the shower is TINY. 2nd bedroom is fine for 2 kids, but not real small ones as they have direct access to the balcony. The family who runs the place is pleasant. Moustafa will answer any questions you have.

For our stay of a week, we had a big appartment (2 bedrooms, livingroom/kitchen), with view over the marmara sea. It was very clean and well taken care off.

At first it was somewhat confusing to me: when I had called for the reservation i was quoted a price with breakfast included. Upon arrival the reception did not know about this, but this was quickly sorted out, and we enjoyed the good turkish breakfast.

Spent 4 nights as a family of 4 with kids aged 10 and 6. Had booked 2 bed apartment but on arrival found it had 3 which was an added bonus. Beds were really comfortable, decor basic but perfectly fine. Little kitchenette was useful. Breakfast was typically Turkish so if you don't want this, better to avoid the €4 each and go out. We were so close to the main sights we did them in 2 days. Transfer to hotel was easy if long. Traffic is shocking in Istanbul. Staff very friendly. Heartily recommend for a family stay.