One thing I love about Tokyo is how much there is to take in while just walking around the city, even when there's no particular attraction to see. The three neighborhoods below are great places to visit without any plans.
Shimokitazawa oozes cool. It's the perfect place to go if you want to spend your day hanging out at little cafes and browsing indie shops for clothes, records, and tchotchkes.
The layout of the neighborhood itself is a little maze-like; you'll probably get a little lost but you also probably won't mind. The area has lots of alleys too narrow for motor vehicles, which means that you can walk wherever you want without worrying about traffic.
Famed for its electronic stores, Akihabara is geek heaven. Other things of note: maid cafes, manga and anime stores, and video game shops. Akihabara Hokoten (pedestrian paradise) is held every Sunday from 1 p.m. until 5 p.m. (until 6 p.m. from April to September).
This is not a common tourist destination, but that's exactly why I went. When I visited, I mostly only saw old Japanese ladies. This neighborhood is not known as "Old Ladies' Harajuku" for nothing.
Old people flock to Sugamo to visit Koganji Temple, where there's a statue that is said to have healing powers. Shops that cater to the elderly line the Jizo-dori street from the train station to the temple, selling clothes and food items at reasonable prices.
The area is also equipped with conveniences for older people. For example, there are plenty of seats along the street and signs pointing to the nearest defibrillator machines in case a heart attack happens.