The "he" refers, of course, to the man speaking to the game player. (Gamma G. is right - in the King James version of the Bible that pronoun, when referring to the Lord, is spelled with a capital.)
However, the "he" in this sentence does not refer to God - nor does it come at the beginning of the sentence, which would naturally require initial capitalization.
This is still part of the same sentence, even though it is not part of the "spoken" part! The sentence goes on, and "he" comes at the end. So it is not capitalized. Your daughter was right to capitalize "You" because it began the sentence, but "he" is correct in the middle. Teacher was right. (Maybe teacher missed the other one you mentioned.)
Now I'll get TOO detailed. Sometimes that helps when you want to make something stick in your mind (and sometimes it'll just drive you crazy). You daughter might have written,
"'You played a great game,' he said. He added, a few seconds later, 'Who made the final score?'"
In the first sentence, "he" is not capitalized, but in the second sentence, it is.
Some comments have noted that there is a problem with "You played that game great." The reason is because "great" is an adjective, and it doesn't describe a verb (action word) like "played." She could use it to describe "game" or even "you" ("You did so well, you great field hockey player, you!")
For the verb "played," she wants an adverb, because - as its name implies - an adverb goes with a verb. It could be, "You played that game greatly," but that would be awkward. There are plenty of other adverbs to choose from.
The exception would be if speaking this way is meant to be part of "his" character, and "he" is shown in the story to make other such common mistakes.
Often, when I'm working on writing, I get to the point where absolutely NOTHING looks right! I second-guess every word, every letter, every punctuation mark. Then I have to get away from it and let my brains unscramble.
