When did the idea of a scriptural "canon" originate?

The best reference for answering this question, is not surprisingly, from Bruce Metzger. He is probably the foremost textual critic of the 20th Century - a fact that both Dan Wallace and Bart Ehrmann would agree with. The UBS4 is, in large part, his baby. His work: The Canon of the New Testament: Its Origin, Development, and Significance can be found online here. Of key interest is the chapter "Attempts at Closing the Canon," which sort of gives a flavor for the process.

Summarizing, Metzger essentially asserts the following key points:

1. Prior to Augustine (c. 400), the Church is less about a "canon" than about a spectrum of the worth of various books. Especially in the East, the key was "grading" the books into those that should be read out loud in church, those that should be read by the ecclesiasticals, those that should be read by layman, and those that were heretical. Those lists were somewhat fluid, but a pretty solid consensus had formed by the mid 300s

   The Latin Church had, in general, a stronger feeling than the Greek for the necessity of making a sharp delineation with regard to the canon. It was less conscious than the Greek Church of the gradation of spiritual quality among the books that it accepted, and therefore was more often disposed to assert that the books which it rejected possessed no spiritual quality whatever. In the search for the highest authority it showed a far more lively feeling for an uncompromising Yea or Nay; a classification such as that of Origen, or still more that of Eusebius, was consequently quite unheard of.

2. Like so many things, Augustine's opinions really held sway for most of the Middle Ages. His opinions on what was "best" effectively closed the canon in practice, if not in dogma

   With Augustine, whose influence upon the Western Church was even greater than that of Jerome, we come to a natural terminus in our survey of debate concerning the closing of the New Testament canon. ... Augustine's treatise De doctrina Christiana ('On Christian Learning' in four books) might well head his works on Biblical scholarship. The greater part of it (i. i-iii. 24) was written in 396-7, but completed only in 426. In ii. 13 he gives our present list of New Testament books (but places James at the end of the Catholic Epistles, thus giving Peter the first place): the four Gospels, fourteen Epistles of Paul, 1 and 2 Peter, 1, 2, and 3 John, Jude, James, Acts, Apocalypse. Although he includes Hebrews in the list (following Philemon) as Paul's, in his later works when he quotes from it he assiduously avoids calling it by the Apostle's name." But while he came to hesitate as to the authorship of the Epistle, he had no scruples as to its canonicity

   Before citing the list of Biblical books, Augustine exercises critical judgement, recognizing that some books are received on weightier authority than others. The Christian reader, he says, "will hold fast therefore to this measure in the canonical Scriptures, that he will prefer those that are received by all Catholic Churches to those which some of them do not receive. Among those, again, which are not received by all, let him prefer those which the more numerous and the weightier churches receive to those which fewer and less authoritative churches hold. But if, however, he finds some held by the more numerous, and some held by the churches of more authority (though this is not very likely to happen), I think that in such a case they ought to be regarded as of equal authority (De doct. chr. ii. 12)".

   Twenty-seven books [in the New Testament], no more, and no less, is henceforth the watchword throughout the Latin Church. Yet it would be a mistake to represent the question of the canon as finally settled in all Christian communities by the beginning of the fifth century

3. The dogmatic interpretation that the received canon was closed was finally brought about in the Middle Ages, nearing the time of the Reformations.

   During the Middle Ages the Church in the West received the Latin New Testament in the form that Jerome had given to it, and the subject of the canon was seldom discussed. At the same time, however, we find a certain elasticity in the boundaries of the New Testament. This is shown by the presence of the Epistle of Paul to the Laodiceans in more than one hundred manuscripts of the Latin Vulgate (including the oldest, the celebrated codex Fuldensis, A.D. 546), as well as in manuscripts of early Albigensian, Bohemian, English, and Flemish versions

   It was not until the Council of Florence (1439-43) that the See of Rome delivered for the first time a categorical opinion on the Scriptural canon.

4. Finally, the Council of Trent (1540s) formalized the doctrines of canonicity and its closedness. Note that Trent is largely seen as the beginning of the "Counter-Reformation" and the thing that creates a "Roman" Catholic Church distinct from the Protestant ones.

   Finally on 8 April 1546, by a vote of 24 to 15, with 16 abstensions, the Council issued a decree (De Canonicis Scripturis) in which , for the first time in the history of the Church , the question of the contents of the Bible was made an absolute article of faith and confirmed by an anathema . 'The holy ecumenical and general Council of Trent' , so the decree runs, '.. . following the example of the orthodox Fathers receives and venerates all the books of the Old and New Testament.. . and also the traditions pertaining to faith and conduct.. . with an equal sense of devotion and reverence (pari pietatis affectu ac reverentia).

Thus, the quick answer to your question is probably best summarized as thus:

• The idea of "canon" originated in the Western (Roman) church, in response to a "gradation" approach in the East.
• Augustine more or less definitively settled what was in the canon for the Nicene Church around 400AD.
• In the Middle Ages and the Reformations, there were minor quibbles around individual books
• Formally the canon itself is best "officially" classified as closed in the 1400s.
• That dogma, however, is more a recognition of actual practice and solidarity than a "new" idea that it was closed.