One of my most peaceful experiences in Istanbul took place at a friend's house in Cihangir neighborhood. My friend's apartment had a balcony that was facing the Bosphorus. The view was peaceful with great skyline, birds cruising over the water, and ships passing by the Bosphorus strait. I would suggest making local connections and try to connect with someone who has a place by the water. If you are going to rent a place, then make sure it has a view of the Bosphorus.
If you are into people watching while relaxing, then, you should definitely check out Ortakoy Mosque. There is a park right next to it, and it's one of the best places to take photos of the Bosphorus and the bridge. I would also suggest taking a long walk by the water, maybe, starting from Besiktas pier (Vapur Iskelesi) towards north, and sit at a cafe once you get tired. You can also go to Uskudar pier, then walk towards Maiden's Tower, and sit on a bench and watch the amazing view. Having lunch at Kanaat Restaurant would be good idea. They have bomb food there! You can find many of the traditional dishes of Turkish cuisine.
Also, I would suggest a great people watching spot: The historical tea garden "Tarihi Cinaralti Aile Cay Bahcesi" in Cengelkoy would be a nice place to bring your own food, and order some tea in the morning. You can watch the locals for hours and enjoy the great view underneath an 800 year-old plane tree located in the garden. Don't forget to order simit, the crispy Turkish bakery/circler bread with a hole in the middle, and covered with sesame seed. Location
