Maybe if you like Japanese sweets, there are lots of long continuing wagashi place. For instance Taiyaki. Try Wakaba in Yotsuya or Naniwaya in Azabujuban. You will be amazed by how delicious a Taiyaki can be!
Another must try might be Monjayaki. It's one of the local foods of Tokyo.
Cuisines from Edo period are also recommended, plus there are leg continuing restaurants. Such as Tempura, Tendon (tempura rice bowl), sushi, sukiyaki, Dojo Nabe etc.
"Best" is difficult...
I think it is authentic Sushi in Ginza or Tsukiji because Sushi is popular in outside of Japan.
I wrote here my recommendations a while ago.