For example, I have heard people say 'remember when we saw that one guy?', or 'my one exam just got postponed'. Is it a form of hedging?
'remember when we saw that one guy?'
I interpret this as meaning "unique". For example, suppose we attended an entertainment event and there were ten players and one of them was wearing a bear's head. Later we might say, "Remember when we saw that one guy - the one with the bear's head?"
'my one exam just got postponed'
I interpret this as meaning "one-and-only". I had only one exam and it was postponed.
It is a determinar used to add emphasis:
One can be used instead of 'a' to emphasize the following noun. [emphasis]
There is one thing I would like to know–What is it about Tim that you find so irresistible? One person I hate is Russ.