Kant’s notion of a unity of apperception seems very useful. The too-easy gloss for this is that it is what properly says “I”. I call it too easy because in ordinary speech, this is typically deeply confused with the empirical, factual “me” for which we also say “I”. Kant rethought personal identity in ethical rather than psychological terms. A transcendental “I” is wholly constituted by a totality of simultaneous implicit commitments.
The term apperception had been used by Leibniz for a reflective apprehension of content. Apperception for Leibniz — in this way like intellect for Aristotle — was neither an intrinisic characteristic of soul as such, nor continuously present in a given soul. It thus seems already to have had an implicitly normative character. We would, I think, speak of a failure to apperceive rather than a bad apperception. Apperception seems to carry a built-in notion of relative rightness for its context. Nonetheless, one Leibnizian monad’s apperception of a given content would not be the same as that of another, due to inherent differences in the total constitution of each monad. To use a Hegelian word, apperception is all about mediated apprehension.
Unity of apperception is a result of the highest of the three kinds of synthesis Kant discussed, formed from a coherence of many apperceptions that are themselves already syntheses. This seems like a more refined, clearer notion of what Plotinus obscurely anticipated in speaking of a sort of microcosmic counterpart of the One in the soul. In Aristotelian terms, unity of apperception is a form and an end. Brandom has particularly emphasized the status of such unity as an ethical goal, rather than something that just factually happens. The movement toward such unity at each moment has to continually try to integrate new content, so whatever unity is achieved is in a way born anew at each moment. I think this process involves something like the free play of reflective judgment in the Critique of Judgment.
The phrase “unity of apperception” also makes it nicely explicit that we are talking about something that corresponds in the first instance to an adverbial expression, and only derivatively to a simple noun. “I” is like that. I have argued that this is generally true of what is conventionally translated as Aristotelian “substance”. (See also What is “I”; Psyche, Subjectivity.)