One of the most famous Tokyo local cuisine is "monjayaki".
Monjayaki is known as another from of Okonomiyaki, but it has different ingredients and broth. It requires instructions to make, but making a monjayaki is also a fun thing to experience! Tsukishima district near Tsukiji is especially famous for monjayaki. There is a street called "Monja Street", with over 100 monjayaki restaurants.
I would definitely recommend "Tendon" which is deep fried seafood/vegetable on a bowl of rice.
If you're in Tokyo, I definitely reccomend "天すけ" (Tensuke) in Koenji where you get to see the Tempura fried and served right infront of you. Go for the Tamago Lunch, you won't regret it :).
However be sure to get there by 11:15 because when I was there a long line has already formed even before the shop is even open!
How about shaved ice (kakigori) ? Recently, there are many shaved ice specialized restaurants have opened, such as Yelo in Roppongi, and ICE MONSTER in Omotesando. Himitsudo in Yanesen area is a long established shaved ice cafe, where hundreds of people line up during summer!
I recommend Yakitori! It’s a Japanese style grilled chicken. You can find yakitori in most Izakaya. It goes with both beer and sake. There are many kinds, made by different parts of chicken body, so try variety of Yakitori yourself!
Sushi is the must of course but if you are more into like local food, try to go to Izakaya which is like a japanese version of tavern. You can experience so many kinds food that most japanese people eat usually with very reasonable price!