I know you're looking for recommendations in Tokyo but Kyoto and Nara are also great cities to explore on two wheels.
Oh alright. Back to Tokyo ... With the sharing scheme, you can pick up e-bikes in just about any part of central Tokyo, so you could get a bus or train to a neighborhood that you want to explore and go pick up a bike in situ.
https://docomo-cycle.jp/tokyo-project/en_index.html
For pleasant areas to explore on bicycles I'd suggestL
Ueno / Yanasen neighborhoods
Iidabashi / Yotsuya / Yasukuni jinja etc. (already proposed above)
Shinjuku Gyoen / Sendagaya / Harajuku / Meiji jingu
Hibiya / Marunouchi / Ginza
Ryogoku / Sumida river / Asakusa
The bike scheme doesn't appear to extend to outer Tokyo wards like Katsushika yet. Perhaps you can find bike rental out there ... It could be quite nice to explore that old-world neighborhood and then ride along the banks of the Edo river.
EDIT: For Shibamata and Edo river you can rent bicycles from Shibamata at 400 yen per day:
http://www.katsushika-kanko.com/rental-bicycle/