Are there "rational" criteria for which books appear in the Bible?

I would also like to point out that during early christian persecution, destruction of christian writings was not uncommon. If, then, some Christian at the time had a large assortment of Christian writings, it would be logical for him to select only the best books for preservation for practical purposes.

I personally believe that this is the process which has preserved most of the NT. The Gospels were considered very important, various apostolic letters, revelation, etc. There was a sort of natural criteria for the canon, since an official one could not have been produced under the circumstances of such persecution. It's resonable to argue that perhaps many other Christian books, both known (ie the lost espistles of Paul) and unkown were destroyed at this time.

In addition to the criteria mentioned above, historically there was a claim of apostolic authorship. See: https://christianity.stackexchange.com/a/5527/1039

This is a difficult question, given that human reasoning is itself flawed. But if we were to look at the canonization process, the most we could say is that it must be not be absurd; Consider how Matthias was chosen to succeed Judas; even though the process of 'drawing lots' seems irrational, the lots were being drawn against a few possible candidates and not merely randomly at large. So the lots were used to make the decision against equally qualified candidates final in a way that no-one could claim favoritism. A very rational choice if there is one.

From my own study (and it has not been perhaps as extensive as many) these seem to be the criteria that explain the process in the most rational fashion:

1. Traditional usage. Writings being added to the rule required that they were traditionally accepted for the use in teaching, admonishing, public exhortation, quotation and so forth. The reasoning here is obvious; this is the primary reason or use for the canon itself: So that a given locality does not have to ask the Bishop who has to ask this other Bishop who has to look at what they're about to read to determine whether its okay to be quoted, to be taught from, to be interpreted as prophecy, or to be announced publicly in liturgies and other prayers. What I mean by this is simply that one may consider this or that book quite true, but if it had no traditional usage, its position would be tenuous. This explains why some other epistles did not get added to the rule; there's only so much time for public exhortation in prayers and liturgies of the scriptures themselves.

2. Not redundant. Some amount of redundancy is allowed, but later letters (if you look at the Apostolic Fathers) contained a fair amount of redundant material, simply re-iterating what was in the Gospels and Epistles of Paul, Jude, John, James and Peter.

3. Has a special position. The writings of St. John had, as far as I can tell, a special position in the minds of the early Church. He was a great hero of the faith, and as we may recall, was the one surviving Apostle. His works - especially Revelation - are given special consideration because of the magnitude of his reputation as a true Theologian.

4. Not praxis. This means, documents which were pure praxis (rules for practices) were excluded such as the Didache. You will notice that there are often problems with Paul's letters where he is 'speaking as a man' - giving practical advice for a given community's situation. Looking back now, we don't know the situation he was speaking to, and thus it is easy to misunderstand his advice. So pastoral letters were mainly excluded, though Paul's seem to have an elevated status (even Peter recognizes this in one of his own letters.) Paul's letters may be pastoral, but they are also theological and prophetic.

5. Part of the Septuagint. Everything in the Septuagint was accepted without question as far as I can tell, since it was the prior canon (what the 'scriptures' meant if a New Testament document refers to them.) The Septuagint - the Greek Old Testament - was the Early Church's canon of scripture.

6. Not pseudopigraphia. This is to say, the author/authorship was known and accurate as far as they knew. So Hebrews was believed to be the work of Paul (which it probably was.) The works attributed to John were either his own words or authorized by him (as with the Gospel of John.) The gnostic gospels were pure pseudopigraphia; faked names against unknown sources.

7. Not written by a heretic. Simply put, some men who entered the faith later recanted; their works would be considered suspect and not included.

So take these criteria and look at The Shepherd of Hermas. Very popular, but never met criteria 1: being read or declared in church. It was a symbolic, poetic work and is difficult to understand. It would fall in the category of something like The Great Divorce, being that it is probably true and perhaps prophetic even, but is the wrong genre to be in the canon entirely.