Why do we use the definite article in the expression “quite the [noun]”?

Question

Like: "quite the singer", "quite the writer", etc. while he/she is just a singer/writer and is not the only singer/writer, etc in that context.

Answer 1

Quite a (and quite the, sometimes used ironically) indicates that the specified thing or thing is recognized as notable, remarkable, or impressive.

Quite a party, isn't it?
It has been quite a year.
Quite the little horsewoman, aren't you?

Quite the thing is a dated way to say socially accepted.

She was quite the thing in heels and stockings and lipstick.

_{[The examples are taken from the NOAD]}

Answer 2

Stating "Quite a [noun]" indicates that one is notable, while stating "quite the [noun]" is remarkable.

It is mostly the difference between the indeterminate (a—or, just another [noun]) and the determinate (the—a specific [noun]).

Answer 3

We use the definite article to show uniqueness.

This case, saying "quite the singer" suggests that the singer is unique in some way -- e.g. being noticeably more talented than the others.

Answer 4

I agree with @awm as in cases where one says quite the [noun], he wishes to indicate a specific trait of the [noun] that even among its commonalities makes a difference.

Answer 5

As noted by @ColinFine, "quite the X" is simply an idiom. It's not any sort of standard grammatical construction and so cannot be explained according to standard grammatical rules. Other idioms that include inexplicable instances of "the" include "what the hell," "the hell you say," and to be "for the asking."