Try Adiguzel Restaurant if you're interested in what the locals eat. No ambiance, but nice and clean, really good food and fairly cheap. Galata House is a restaurant run by a friendly couple, formerly a prison, definitely an interesting place to see, nice Russian/Georgian dishes but not too many choices. Damat Pacasi, nice little restaurant, a good variety of filling food that you can watch as they're cooked.
All are a minute's walk form the Galata Tower.
That's an awfully large area, but I can narrow it down for you. Don't eat in Sultanahmet. It's best to avoid any restaurant in that area as they are low quality Turkish food (in general) for people who don't know much about it. In Galata, you've got a lot of choices for places- my favorite is the Galata fish house, Furreyya. They do a mean grilled octopus and a fantastic Sea bass (Ahtopot and Levrek, respectively).
Other than that, if you feel like heading up the hill from Galata a bit, you can get to Asmalimescit, which is the neighborhood parallel to Istiklal Cad. closer to Tarlibasi. There's a bunch of fun Meyhanes, tons of bars, and lots of local foodies eating everything.
Hello there. If you would like to eat great Turkish food, drink some drinks, and listen to amazing live Turkish music, Badehane is the place. Badehane is located in Galata, right around the corner from one of Istanbul's best music venues called Babylon. It is a little pricey, but every Wednesday night they host live music, starting at around 9:00pm. Also, there are plenty of restaurant/bar options in that neighborhood.
Hope this helps...