I'm a politically engaged person, so forming a philosophical foundation for my political beliefs is quite important to me, and will serve as a guide for my political opinions and activities. So, I'm going to be looking at a few different such foundations.

• Right off the bat, I'm going to rule out any political philosophies that are founded on fallacious appeals - appeals to nature (anarcho-primitivism), appeals to tradition (religious fundamentalism, divine right of monarchs), appeals to authority (totalitarianism), etc. These sorts of appeals are not generally accepted as valid logic, since they rely on unsound predicates - all that is natural is good, all that is old is good, and so on. These are poor fits for a foundational political philosophy.
• Liberalism: In this context, liberalism (as advocated by Mill) refers to the idea that the primary goal of society to protect individuals' rights, thus maximizing liberty. This is great and theory, and in practice it essential to safeguard some liberties, but there is also a sense in which this is an impractical way to run a society. Very often, different "rights" are in conflict. We might suppose that both life and property are fundamental rights; but then what if we're considering a universal healthcare system? One person's right to life might dictate medical treatment, even if they do not have the means to pay for it; but another's right to property might dictate that they not be taxed to fund such a system. Furthermore, a system that adheres perfectly to this form of liberalism will lead to undesirable outcomes, such as massive income inequality.
• Utilitarianism: Mill had another theory: utilitarianism. This is the idea that a society should seek to do the most good for the most number of people; to maximize the sum of the utility function over the entire population. It certainly sounds reasonable, but in certain situations, it seems to break down into unacceptable results. The classic example is that of 5 patients who need organs or else will die, and 1 healthy man. Would it be okay to kill the healthy man for his organs, and use them to save the other 5? According to (act) utilitarianism, yes. (Interestingly, this conflicts with Mill's liberalism - I don't know if he ever addressed this conflict.) Rule utilitarianism fares no better, as the general rule not to lie (which would usually increase utility, since it builds trust) would break down if a murderer asked you for your family's location.
• The Social Contract: Rousseau viewed government as a "social contract" - a metaphorical agreement between individuals in a society and their government, whereby the individuals give up some freedoms, in exchange for protection and the respect of others. Social contract theory was also supported by Hobbes, who saw life without government as, in a word, terrible. While I think there is some value in social contract theory, it provides little basis for actually deciding on policies. Rousseau argued that the government is responsible for serving the "general will" (what's best for society), rather than the "will of all" (what the people want the government to do). This is problematic, because it means social contract theory can justify all sorts of tyranny under the broad category of the "general will." Hobbes didn't even try to hide that facet - he was openly an absolutist. So it's a nice idea, but a poor basis for a policy.
• The Original Position: John Rawls presents another alternative: the argument from original position. The idea here is that society should ideally be what rational people in the "original position" would agree on - meaning that they are deciding on what society should look like without knowledge of what position they will hold in that society. Rawls then argues that such people would be risk-averse, not willing to gamble their lives on being part of a desperate minority. From there, they agree on two principles: first, all individuals would be guaranteed basic human rights; and second, the society would seek to equalize opportunities, and if there are any inequalities, they would benefit the least advantaged people. This is the "maximin" strategy from game theory, and I think its application here is ingenious, as its effectiveness has been experimentally verified in many situations, including Chess.

Rawl's argument from original position is very satisfying to me, as I can't think of any problematic examples like with liberalism and utilitarianism. If liberalism's problem is only considering rights, and utilitarianism's is only considering consquences, then the original position strikes a decent balance. Of course, even it is not perfect, since it can often be rather vague and hard to apply. One problem is that you need to define what basic human rights are, and another is that it is hard to determine whether an inequality is "acceptable" under the argument. But, in any case, it's a good starting point, and it's not as obviously flawed as the alternatives. When in doubt with the original position argument (which rules out rights violations), I'll probably fall back to utilitarianism as a "tie-breaker."