According to Publius (probably James Madison), factionalism is bad because factions, including factions that constitute a majority of the population, can act in their own factional interests in ways contrary to the national interest. But what precisely is this national interest against which factions can act? Since Publius explicitly talks about factions in the majority (minority factions are countered simply by the basic majoritarian structure of the government), clearly a "factional interest" cannot be the interest of the majority, at least not in any simple sense. Similarly, just that factions do have interests opposed to the national interest means that the national interest cannot simply mean "in the interests of everyone in the nation;" if it did, then no faction would have interests to the contrary. None of the simple answers work, so we must look to more complex explanations of this mysterious "national interest" that we must protect by carefully structuring our government.
The moral and political zeitgeist at the time of the American Revolution and the creation of the US Constitution a decade later favored an "objective" view of morality and political rights. I mean objective in a sense similar that scientific truths can be known by a reasonable method, both ontologically and epistemically: these natural rights are actual properties of individuals, and they can be known by some reasonable method.. The language of the Declaration of Independence unequivocally supports this interpretation: "We hold these truths to be self-evident..." establishing epistemic objectivity, "...that all men are created equal, that they are endowed by their Creator with certain unalienable rights," establishing ontological objectivity. Opinions about rights are as irrelevant to their existence as are opinions about the law of gravity: rights are properties of individuals that exist independently of our notions of them, and their existence and nature are rationally determinable.
But there is more to government than simply establishing these rights. If that was all there was to government, then there would be no real need for elections. We would be better served by creating a more-or-less meritocratic system with economically disinterested agents acquiring expertise in determining and implementing these objective rights. In other words, we would organize our government as we organize the scientific community. But there are other decisions we need to make socially, decisions for which we can find no better justification than the will of the majority of the people. These decisions come into the purview of government not because they are of the same character as natural rights, but rather because the tools of government (i.e. coercion) must be employed to implement those decisions. (Anarchists and Libertarians would probably disagree, but that's a debate for another day.)
It's clear, then, that the Constitution exists to manage the tension between these two functions of government: to carry out the popular will where legitimate (in those spheres where coercion is necessary to do so) while still protecting the objectively existing natural rights of individuals and minorities against the illegitimate popular will. The men in Publius's government must keep one foot in the popular realm and another in the realm of the ideal, the basis of natural rights. If Madison et al. were correct about this foundation of individual rights, we could hardly fault them. While not the only possible solution, the structures in the Constitution seem difficult to actually improve upon. But if he is not correct, then considerable scope for improvement becomes possible.