Cihangir's changing so rapidly it's hard to suggest a particular shop or restaurant that will still be there a few weeks from now. A few that have (so far) stood the test of time:
One of few green places in Istanbul, this Cihangir garden is incredibly picturesque and a wonderful escape from the business of the city. The outdoor restaurant is set amongst trees and feels like someone’s well-landscaped back-yard. Also a great spot for brunch.
This was once Istanbul's local gay hangout and it remains one of the most fabulous places in town. The decor is wonderful - red walls, purple chairs and fantastic chandeliers inside, and an outdoor terrace with less exciting decor but stupendous views above (only open seasonally). I’d call it more campy than upscale, but it remains one of my favorite places in town, and the prices are more reasonable that some other similar venues.
A streetside Cihangir bar with a 70's ski-lodge atmosphere, and the best free cerez (nuts) in town. A little on the pricey side but recognized for delicious fare and the opportunity to watch the Cihangir locals sipping cocktails.
Much has been made of this local breakfast joint, which serves fresh cheese and fruit in a style attributed to Van, a city in eastern Turkey. The simple, delicious, and reasonably priced fare is draw enough, but the friendly (if sometimes frazzled) waitstaff is what keeps locals and tourists alike returning day after day. I once had houseguests who went to Van every single morning of their two week stay. Come early as there is a line at peak times.