"Buy her that one thing she won’t buy for herself."
I noticed "...that one thing..." is a very special usage here, where as a non-native speaker I would make a sentence almost same (but without "ONE")". So, I would simply say "Buy her that thing she won’t buy for herself"
But, I wonder why the writer might have used (...THAT ONE THING... rather than "THAT THING"). Do you think he did it;
a) because he wanted to mean "If there are many things that she couldn't afford, buy only one such a thing for her, but one is enough so buy one, NOT two, NOT three, not FOUR etc.",
b) or rather than the number of the things, he wants to refer to the "degree of her LIKING. The writer may have wanted to use it "THAT ONE THING" which people commonly use when pointing at the things they liked at the shop windows, eg "THAT ONE on the right, THAT ONE over there". So, maybe the writer used it to refer to the MOM's possible sentence when pointing at "THE PRODUCT" she liked most. (when she was out looking at the windows, she said she liked THAT ONE the most and you asked her "which one?" and she pointed at it saying "THAT ONE"? May be the writer wanted to quote her.
So, really, why "THAT ONE THING" instead of "THAT THING"?