Which areas in Tokyo would you visit?
[Hotel in Tokyo]
We originally booked the Hilton Tokyo but there was a TV programme on in the UK a couple of weeks ago called the busiest train station in the world and it focused on Shinjuku Shinjuku Stations are
not the closest stations to Hilton Tokyo.
You can choose Nihi-shinjuku Station (M-07) or Tochomae Station (E-28). Each of these two subway stations is much simpler and much smaller compared to the Shinjuku terminal. None of the two is a station for a transfer to/from another line.
The mentioned programme, I presume, focused either on only Shinjuku Station of JR-East or on Shinjuku Stations of five operators as one terminal.
Anyway, if you start from Hilton Tokyo, you can visit many tourist spots in Tokyo neither embarking nor disembarking in Shinjuku Station(s).
For your information: Tokyo Metropolitan Government Buildings are located nearby. Building No. 1 has two observatories about 200 meters above the ground, which do not require any admission fee.
http://www.metro.tokyo.jp/ENGLISH/TMG/observat.htmWe also have the Hotel Metropolitan Edmont in Iidabashi booked for the same period. We have stayed here before or 4 nights and we had no problems with transport. Do you remember which line(s) you used from Iidabashi?
Three railroad (including subway) operators have Iidabashi Stations on five lines. That may be both a merit in terms of convenience (though it depends on your destinations) and a demerit in terms of a risk of a small kid getting lost.
In Iidabashi, platforms for subway Tozai Line and JR Chuo (Sobu) Line are rather easily accessed from Hotel Metropolitan Edmont, while platforms for the other three subway lines are a little away.
If your comment "we had no problems with transport" is about all the five lines, Iidabashi may be better.
However, if you have not used the further three subway lines from Iidabashi, I am not sure whether the same comment will apply to them.
Anyway, please note that convenience concerning railroad (including subway) in Tokyo may depend highly on which route you choose, not only on which area you start from.