If you look, you can find many "equivalent", sufficient axiom sets for classical propositional calculus. The set {CqCpq, CCpCqrCCpqCpr, CCNpNqCqp} seems like the axiom set most commonly used. There do exist systems with only one axiom, and systems with no axioms, so why is {CqCpq-1, CCpCqrCCpqCpr-2, CCNpNqCqp-3} more common than other axiom sets? Does there exist a rational ground as to why one would select this axiom set over any other given axiom set? I can see why one might select it over axiom sets with just one axiom, as those axiom sets at least very often involve a long axiom which also doesn't seem very intuitive, or at least I believe that most people would not find those single-axiom systems as all that intuitive. But, this in no way guarantees this particular axiom set as more intuitive than any other sufficient axiom set, given that one grounds the the selection of this axiom set over any other via "intuitive" considerations. Also, that the axiom set above comes as all that intuitive in comparison to at least one other axiom set seems doubtful (see final paragraph).
What follows indicates what actually prompted this question in my mind. Say we compare the axiom set {CCpqCCqrCpr-4, CCNppp-5, CNpCpq-6} with the above. Why prefer the first set over the second in light of that 1. Jan Lukasiewicz, the logician responsible for both axiom sets as sets (not the axioms, the sets as sufficient and non-redundant for classical propositional calculus), at least appears to have preferred the second axiom set to the first (as evidenced by his book on Aristotle's Syllogistic, and his textbook Elements of Mathematical Logic which uses the second axiom set, but not the first) and 2. the second axiom set is simpler than the first axiom set in the sense that the total number of letters is smaller for the second set than for the first set? The only answer I've come up with so far, other than historical accident, comes as that if you have the rules of conditional proof along with substitution and conditional elimination, the first set works out as simpler in that it that 2 of its axioms can get eliminated because of the rules of conditional proof, while the second axiom set comes as more complicated in that only 1 of its axioms can get eliminated.
The question "Why is this set {CqCpq, CCpCqrCCpqCpr, CCNpNqCqp} preferred over any other sufficient axiom set?" seems a generalization of "why prefer the first axiom set over the second?", which I think indicates why the particular question of "why prefer the first axiom set over the second axiom set" seems relevant to the general question here.
As a possible pre-emption:
If one tries to argue that the first set involves more intuitive axioms, I simply don't see that. It seems at least plausible to argue that "most" people with experience with logical reasoning before learning formal logic would probably not find 1 intuitive (since it's a "paradox" of material implication), 2 does not seem intuitive either, and 3 seems at best somewhat intuitive (it doesn't even directly correspond to the rule of modus tollens as commonly presented, since you have CNpNq as the antecedent, instead of where you have Cpq as a premise for modus tollens, and a negation Nq as the other premise. This at least seems to imply that if you interpret it in the light of modus tollens you'll have to assume that NNp==p to understand the axiom, making it not all that intuitive). On the other hand, I would expect that 4 comes as one of the most intuitive, if not the most intuitive formulas of formal logic. So, even if 5 (which I don't find intuitive) is not intuitive, and 6 is not intuitive, the second set as a whole works out as more intuitive than the first set.
Edit: related question at math.stackexchange.