We know an expat that's currently living in Tokyo and this is her suggestions on what to bring/buy from your home country as things are very expensive there, especially if you live in the foreigner district. Local will be cheaper.
Nappys/diapers
Wet wipes (the ones here are much thinner and not as wet as the ones you get in the west)
Medicine (ie cough medicine, flu medicine, paracetamol (for adults and children) aspirine etc)
Baby Food, here it is difficult to get oatmeal so if the child eats it then i would suggest to try and bring some.
Baby Snacks (biscuits etc). Here you can get but everything is very much based around rice
Deoderant
Shampoo and conditioner (no problem to get here, but its very difficult to see which is which)
Shower gel/bath gel (same as above but as yet i cant see to distinguish which is bath and which is shower)
Feminine Hygene products
Toothpaste/Toothbrushes
Cleaning products for the house, dishwasher detergent, Laundry detergent, if you use whitener/stain remover for the clothes definately bring some as they dont seem to have it here (well i havent found it yet). Washing machines dont have temperature setting so it is difficult to remove stains without scrubbing before putting in the washing machine.
Tea (very expensive)
Coffee (coffee is very expensive here so i would recommend to definately put a stock in the container)
Bed Linen/pillow cases (the beds here in Japan may differ in size to back home so best to pack some spare sheets and covers)
Breakfast cereal, except for in National Azabu choice is very limited and very expensive
Pasta
If you have special baby bottles, dummys and equipment for the baby best to try and bring a stock also as here they dont have western brands.
You can get of course everything here but as mentioned it is very expensive and also when you first arrive you dont want to have to be worrying about these things. Once you get more settled you can then spend the time to look at what is what and where to buy.
Possible items to bring to Tokyo from home country
Dear yippy,
First of all, welcome to Expat-blog! 
And thank you so much for this contribution which I'm sure other members, seeking to move to Tokyo, will find extremely useful.
If you have spare time and would be interested, it would be really nice if you could bring similar contribution by posting articles in the Life in Japan guide.
Enjoy your stay with us! 
Take care,
Haf
Hi Yippy
I raised my son for the first 4 years of his life in Japan and I recommend buying the Japanese brands of diapers. They are a good price and excellent quality.
Regarding wet wipes, the Japanese ones are definitely thinner so I always purchased mine from Costco or The Flying Pig (online shopping from Costco including delivery). The brand is called Kirkland and they are thick and moist.
Regarding the Japanese baby snacks, it is true they are often rice based but there are some really great ones your baby will enjoy.
Don't bother bring shampoos, conditioners or body soaps. They have tons here including a lot of foreign brands. You can always also buy foreign ones from Costco.
Feminine hygiene products vary. For all types of pads/napkins the Japanese ones are really good. From memory, they have a very limited range of tampons, but do have the Tampax applicator ones though only the light and medium sizes.
Cleaning products, they have plenty that work just fine, and there is always Costco.
Honestly don't remember thinking tea or coffee were expensive.
Breakfast cereal is a problem. Mostly sugary cereals though the foreign stores, Costco, etc. have some decent muesli. If you absolutely must have your cereal, then bring a few boxes until you find a suitable replacement.
Pasta. That's a funny one. I remember reading that the Japanese were the biggest consumers of pasta in the world. Regular supermarkets have a decent selection of pasta. Kaldi stores which sell foreign goods, coffee and sweets have very impressive pasta varieties. You definitely don't need to bring pasta.
Check out japanfamilyguide.com for details for some of the shops I mentioned plus other family-related stuff.
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