Question: I can't remember when and where I had this discussion, but I remember being corrected when I was speaking by a stranger saying that it is never correct to say *give me half of this*; instead, the grammatically correct phrase would be *give me one half of this*. I've never been a pro at where numbers fit in with the English language, so maybe someone here could shed some light on this.

Answer: 
I had this situation recently. It was so embarrassing when the saleslady told her co-workers about what I said. 
I asked her to give me one half kilo (1/2 kilo) of prawns and 3 pieces of fish. She ended up giving me one and one half kilo of prawns. When she asked for the payments and I asked the amount, i freaked out. I was not expecting that she actually gave me 1 1/2 kilo of prawns. 
I told her I asked for one half not one and one half kilo. She got angry and her co-workers started to laugh. They were all telling me I was wrong. They said it's half not one half.
In my understanding, "one half" and "half" are both correct it depends on how you use it with other supporting words.
If you are discussing arithmetic, it is always correct to stipulate a numerator and a denominator. "Give me half an apple" is fine, but "what is half plus a third?" is incorrect and should be "what is one-half plus one-third?"