Recommendations - How to spend your extended stay in Tokyo
If you are staying by yourself then suggest to sleep over at a local Capsule hotel. Capsule hotels are unique invention of japan namely for the weary office workers ("salarymen") who travel long distances daily or for those looking for a cheaper and cleaner alternative for a good night`s sleep. If you are up for the challenge, go with an open mind and spend a night or two at a Capsule hotel. Guaranteed, it will be one of your most interesting takeaways from your extended stay, and something to boast to your friends back home about your unique experience. Friendly caution: Do readup on wiki about Capsule hotels before visiting one. Places recommended: Daimon.
If you are with family and looking for unplanned but exciting adventures in Tokyo, then take Tokyo Metro One-day Open ticket, get on any of Metro trains, get off any unfamiliar stations and just walk around. You can always ask station staff for directions, popular places to visit in that area, good places for lunch or dinner. Don`t forget to collect Tokyo Subway Map before embarking on your adventure.
If you wish to experience hot springs (onsen in Japanese) but dont wish to stay overnight, hop on Tokaido train (from Tokyo station) going to Izu which is couple of hours from Tokyo station.
Another experience worth taking is visiting Boso Peninsula, which is the farthest corner of Japan in South East direction. As much as you can enjoy fish and hiking, the most exhilarating experience is to stand at Nojimazaki point and seeing endless ocean ahead. You gotta see it to believe it.
If your stay is extended to few weeks than days then checking out Kyoto, Nara and going as far as Nagasaki is recommended.