Vacation rental in Tokyo - help with neighborhood needed!

Nishi Ogikubo (nishiogikubo on hyperdia.com) is 13 minutes from Shinjuku station and 27 minutes from Tokyo station.

Both Nakano and Nishiogikubo are on JR Chuo commuter line, so handy for transport. As for the neighbourhoods are concerned, they are quite popular residential area with commuters, with a good range of amenities, but of no spercial interest to tourists. Ideal if you just want to blend with locals - without particularly interesting or exiting things to do or see. Beware of overcrowded trains during weekday rush hours - 7.30 to 9.30 am into central Tokyo, and 4.30 to 6.30 pm in the opposite direction. I'd not be happy travelling with a one-year old then, with a stroller etc. For that reason, I'd look for somewhere more central.
In that price bracket, there is a huge range of apartment rentals available right in Central Tokyo within JR Yamanote loop line, where you want to be. Look at Oakwood Mansions in Azabu (near Roppongi), and Weekly Mansions - just a studio appartment with kichenette, but costs only around 12,000 yen a day for a triple.

Some firsthand experience to share here, little_berlin (assuming you check back at some point).

I stayed in one of the 1 bedroom units at Nakano from this agency in January of 2009, and generally liked it. Note that a 1BR in Tokyo is really more like a studio, in that the BR is the living room during the day. The kitchen, on the other hand, is entirely separate. These units also have their own washing machines (mine was on the balcony, a pretty common arrangement).

The apartment is a few minutes walk from the train/subway station (very, very convenient), and to get there you walk first through the quiet residential area and then through a very typical shopping/entertainment district. Lots and lots of small restaurants catering to locals, and there's a good grocery store nearby. Although trains were busy at rush hours I didn't find it particularly daunting and I'm even claustrophobic---the fact that at 5'6" I'm taller than most people on the trains meant that I didn't have as much trouble as I might have otherwise. If you take the subway line to another point on the Yamanote (instead of the Chuo to Shinjuku) it's even less of an issue.

Note that winter is pretty chilly and damp in Tokyo, and my apartment did not have central heat, instead relying on a space heater to heat the living area/BR (and hot water, of which there was plenty as it's an in-line heater, to heat the bathroom). You'd all three be sleeping in the single room, and it's not huge.

The advantages would be the more complete and larger kitchen (as opposed to the kitchenette in a weekly mansion) and the in-property washing machine (very handy with a toddler).

I will say that I very definitely got the feeling that I was "living Japanese", right down to compulsively sorting my trash and recyclables. The agency does a very good job of making sure that you're met at Nakano station, and of orienting you to the neighborhood, etc. Everything went very smoothly from start to finish.

I would still trade in a washing machine and a full kitchen with central location, with a choice of short subway rides or taxis to get to and from major sights and shopping/entertainment districts.
Remember, you are on holiday in Japan and you just don't want to spend any time mixing with daily commuters and staying in pleasant, but completely uninteresting suburbia. Probably the agency knows how confusing it is for guests to find the apartment block, hence the need to escort them from the train station. In Central Tokyo, just show the address to the taxi driver, they will put it into their sat nav and you are brought almost to the front door.