The notions of logical consistency, pertinence and possibly relevance seem important for a philosophical inquiry on some matter to be fruitful and interesting.
Other notions once considered important, such as the notion of causality, are possibly irrelevant and even inadequate in certain contexts.
I wonder if it would be possible to mention some highly frequent unifying characteristics or general principles of most historically relevant philosophical theories and proposals?
When it comes to political philosophy, one unifying challenge is to describe the position of every person in the society once the new philosophical system is in place. It is common for a philosophical work to go on and on about the benefits that will fall out of the sky for certain favored groups, and ignore completely the groups that will shoulder the burden of providing these benefits. Maybe the disfavored group deserves its loss (as in the example of masters losing their rights to enslaved persons), but the philosopher is obligated to say that out loud and to explain why.
I would say the unifying principle is what Plato called dialectic. Roughly, the social practice of giving and asking for reasons while seeking the truth.
Note that I said "seeking the truth" above to avoid the implication that philosophical dialectic will ever terminate in some final perfectly true theory. In fact, the Socratic roots of philosophy indicate that the wisdom gained from engaging in philosophy is coming to the realization that one likely will never know anything with certainty (epistemic fallibilism). That is fallibilism is not a prior principle of philosophy, but a shift in existential stance that may result from the practice of philosophy.
Many of the other principles mentioned by the original poster and in some of the comments are views articulated within philosophical discourse rather than being framing conditions of philosophy itself. Even the principle of non-contradiction is put forward and defended as a philosophical thesis in Aristotle, and certainly there are philosophical texts both before and after Aristotle that have challenged the truth of the principle of non-contradiction (dialetheism) although this is a minority position.
One could say that some Continental philosophy challenges the idea that philosophy is a game of giving and asking for reasons seeking truth, and instead reframing it as a play of texts (Derrida), a form of artistic expression (Deleuze), or a raw power struggle (Nietzsche, Foucault). However, I am unconvinced by this. I would say that in each case, these authors only succeed in shifting the genre of their thought from philosophy to something else. One can argue that we should stop doing philosophy and instead engage in these other kinds of thinking or text-generating activities, but that is not an argument against philosophy itself being essentially what Plato called dialectic.
