4 ideas
A person is the same person over time exactly as far as their consciousness can reach back to past thoughts and actions. The self is not fixed by the underlying substance, soul or body. It is fixed by which past experiences a present consciousness appropriates as its own. Where consciousness does not reach, the person does not reach either.
Whether something is 'the same' can only be settled once you specify what sort of thing you are tracking. The same mass of matter requires the same particles. The same plant or animal requires continuity of an organized life, even as its particles change. The same person requires continuity of consciousness. These three criteria can come apart, so one object can be the same man but not the same person.
'Person' is a forensic term: it names the bearer of praise, blame, reward and punishment. It applies only to intelligent agents capable of law, happiness and misery. Because accountability follows consciousness, a person can justly answer only for the actions they can own as their own. Personhood is therefore a moral and legal category as much as a metaphysical one.
Locke asks whether a drunk man who cannot recall his act is the person who did it. Human laws punish him with a justice suited to their way of knowing, because the want of consciousness cannot be proved and the fact is proved against him. At the Day of Judgment, he holds, no one will be made to answer for what they know nothing of, since God knows every heart.