As has been pointed out, the overwhelming form of the idiom is the blink of an eye. So there's no issue of correctness involved. The questioner, however, had some specific questions that deserve attention, since they suggest some underlying grammatical misunderstandings.
In this context, doesn't it make more sense to use the indefinite article "a" instead of the definite article "the", since there can be more than one "blink of an eye"?
This is not a function of the definite article in many situations. For instance,
- We dialed the wrong number,
- *We dialed a wrong number
even though there is only one right number, and millions of wrong ones, it's always the wrong number.
Articles, like other syntactic particles, don't really have any meaning; they're just a convenient set of labels to attach to just about any set of things we might want to distinguish from one another. They have lots of syntactic functions, though: for instance, a predicate noun has to have an indefinite article, as well as some form of be:
- *He is doctor.
- He is a doctor.
- He is the doctor.
Only the second sentence above is a predicate noun construction. The first is ungrammatical; and the third is an equative construction, with the doctor referring to some previously mentioned doctor (or, alternatively, to some social role he is acting out), but not necessarily predicating Doctorhood of him.
Is "a blink of an eye" incorrect in this context?
No, it's just rare. This question's been answered.
However, one should be careful about using the term "correct" in talking about grammar, especially English grammar, which is mostly syntax, and most especially when dealing with syntactic phenomena like articles. Most ideas about "correctness" (and those are scare quotes) come from vague generalizations, while a great deal of fact is actually known about article usage in English. There are dozens of special uses for articles -- an applied linguist once told me he'd counted more than sixty -- and they don't make much sense at all.
Why, for instance, is it The University of Michigan and not *The Michigan State University? Or The Missouri River and The Nile River, but not *The Lake Superior or *The Loon Lake?
Are idioms like this exceptions to normal definite and indefinite article usage, even though the literal meaning of the idiom makes better sense otherwise?
No, not really. There is no "normal definite and indefinite article usage" in terms of "making sense", which is a semantic concept involving meaning, not a syntactic concept involving grammar. Grammar has nothing to do with making sense; grammar has to do with constructions and how they are used.
Moral: Don't confuse names with descriptions. What's called "the definite article" isn't necessarily more "definite" (and note what a slippery concept that is :-) than anything else; it's just one more syntactic marker, like the to that marks an infinitive, or the to that marks the direct object of listen, and no more meaningful by itself.