In contrast to the answer by Recurse,
This case would be an irregular usage of the
Not really irregular at all; the just signifies that the speaker has a specific instance of the set in their mind (or a specific subset, in plural). "The current US president" is a specific instance of the set of US presidents; "the soup of the day" implies there is exactly one soup being advertised as "soup of the day" in the salient context. "I'll have the coke" means the one single coke you see in your friend's fridge when everything else is beer. None of these need a prior mention, just a reasonable expectation that the particular single instance would be recognised by the listener.
Thus... I'd judge "Nefertiti is famous for the hat" as strange: if you need to tell me what she's famous for, then I likely don't already know it, and couldn't recognise which hat you were referring to.
"Nefertiti is famous for the hat she always wore" is okay, because I now have the specific instance of a hat in my mind: the (single) one that she always wore, as identified by those words.
"Nefertiti is famous for her hat" is also okay, with a similar meaning.
I would think all of these considerations apply to the Jeopardy-like guessing game as well, and I would not think "This person is famous for a hat" a good hint.
(Completely irrelevantly, if I heard "This person is famous for a hat" on Jeopardy, my answer would likely be "What is the Sorting Hat?", because some hats have personalities, too, and get famous despite being just a piece of clothing.)
But rephrase a bit, and things change. Without the pragmatic requirement of "famous": "Here's Nefertiti's bust, of course she's wearing the hat." is perfectly fine, as this sentence presupposes that the listener would know which hat is relevant to Nefertiti.
Also, the OP's "This person is famous for wearing a hat" is perfectly fine, as we do not need the listener to be able to identify which hat we are talking about.
EDIT: CGEL, Chapter 5, Section 6.1 (emphasis mine):
In general, use of the definite article presupposes the existence of the entity, set or quantity that the addressee is expected to be able to identify.