Hiking near Tokyo
Below is a quick run down of locations for hiking near Tokyo but see the region's page for more about the region concerned.
Tanzawa range
The Tanzawa range is pretty much the most accessible mountain range close to Tokyo - especially for those foreigner friendly zones of Shibuya-ku or Minato-ku. You can take the Odakyu line for about an hour or so to [Shibusawa] or [] stations and then take a bus from there for a day hike. The trains are pretty frequent and as the Odakyu line starts at Shinjuku you can pretty much guarantee a seat on the way out and have a good chance of getting one on the way back as well. There are also a couple of hikes that are good for children, with onsens at the end as well.
On the other hand there are longer hikes you can do as well, with Tanzawa itself being one of Japan's "100 Famous Mountains" but requiring an overnight at a mountain hut.
Kamakura
Kamakura is better known as a sightseeing destination South of Tokyo and pretty accessible, with its "Big Buddha" and enough temples to shake a large bamboo pole at. However there are a couple of very pleasant and very historic walks around the hills on which the town is built that are also suitable for children and allow a chance to see the town afterwards.