What are some good classic ideas of what to take home from Tokyo and Kyoto?

If you’re a home chef like me, then Japan has the highest quality knives in the world. I already owned a Shun knife set before my two trips to Japan in 2013 and 2017. But on those trips I picked up a beautiful yanagiba (sashimi knife), deba (whole fish preparation knife) and smaller santoku. In Tokyo’s KitchenTown and along the streets in Kyoto, I bought soy sauce bowls, ramen bowls, chopstick rests, etc. And in Kanazawa I bought beautiful little dishes ornamented with gold leaf.

But it even better than all of those things, which I use regularly, are my photos of this incredibly beautiful country.

Expanding on the 'blade' theme: I also bring back scissors and gardening tools. Some of the Silky saws I pre-order from Rakuten. Food: tea in pretty cannisters makes nice omiyage. I've been asked to bring back yuzu syrup. Japanese cosmetics are also frequently requested. I hit the shrine sales for old textiles and shikishi -calligraphy or painting on square boards - lightweight, easy to carry, one of a kind. I also like Nakano Broadway & similar for anime cels. This is all very personal of course!

I found some delightful souvenirs and gifts (e.g., origami paper, various silk items, etc.) at the Kyoto Handicraft Center. For more information about that center and other shopping options in Kyoto:
https://www.japan-guide.com/e/e3958.html

And for info on shopping in Tokyo:
https://www.japan-guide.com/e/e3053.html