What is it that pass from this life to the next?

How would you answer this question so that it can enlighten/delight/pacify someone from a non-Buddhist tradition

Notions of 'rebirth' are not considered related to enlightenment in the older scriptures. Also notions of 'rebirth' cannot lead to peace (pacification). The scriptures say notions similar to 'rebirth' have effluents (ignorant mental states), result in attachment/acquisition however can have the benefit of promoting merit/morality.

And what, bhikkhus, is right view that is affected by the taints, partaking of merit, ripening in the acquisitions? ‘There is what is given and what is offered and what is sacrificed; there is fruit and result of good and bad actions; there is this world and the other world; there is mother and father; there are beings who are reborn spontaneously; there are in the world good and virtuous recluses and brahmins who have realised for themselves by direct knowledge and declare this world and the other world.’ This is right view affected by taints, partaking of merit, ripening in the acquisitions.

MN 117

The scriptures clearly unambiguously say it is the same person, the same man or woman or the same being that is 'reborn' according to their kamma.

Here, student, some woman or man is a killer of living beings, murderous, bloody-handed, given to blows and violence, merciless to living beings. Due to having performed and completed such kammas, on the dissolution of the body, after death, he reappears in a state of deprivation, in an unhappy destination, in perdition, in hell. If, on the dissolution of the body, after death, instead of his reappearing in a state of deprivation, in an unhappy destination, in perdition, in hell, he comes to the human state, he is short-lived wherever he is reborn. This is the way that leads to short life, that is to say, to be a killer of living beings, murderous, bloody-handed, given to blows and violence, merciless to living beings.

MN 135

The famous Jataka Tales/Past Life Stories of the Buddha refer to the same person & people being reincarnated from life to life.

Suttas such as AN 3.15 report the Buddha saying about his past lives:

Now, monks, the thought may occur to you that the chariot maker on that occasion was someone else, but it shouldn't be seen in that way. I myself was the chariot maker on that occasion. I was skilled in dealing with the crookedness, the faults, the flaws of wood. Now I am a worthy one, rightly self-awakened, skilled in dealing with the crookedness, faults, & flaws of bodily action; skilled in dealing with the crookedness, faults, & flaws of verbal action; skilled in dealing with the crookedness, faults & flaws of mental action.

Suttas such as MN 143 refer to the same (former householder earthling) person returning to earth from heaven as a god/angel to talk to the Buddha & The Venerable Sariputta.

"That is what the deva's son said. And [thinking], 'The Teacher has approved of me,' he bowed down to me, circled me three times, and then disappeared right there."

When this was said, Ven. Ananda said to the Blessed One, "Lord, that must have been Anathapindika the deva's son. Anathapindika the householder had supreme confidence in Ven. Sariputta."

"Very good, Ananda. Very good, to the extent that you have deduced what can be arrived at through logic. That was Anathapindika the deva's son, and no one else."

The doctrine of 'anatta' ('not-self') is not related to the doctrine of 'kamma'. When the Buddha taught the doctrine of good & bad kamma to unenlightened people the Buddha never mentioned anatta (not-self).

In summary, the Buddha always taught "a being" or "person" is "reborn" (example SN 42.3). If oneself was not assumed to be "reborn", an individual would have no incentive to do good & make merit.

In conclusion, we should not consider Buddhism to be special or always enlightened and different to the other superstitious religions. Buddhism is saturated with superstition. Therefore, when we try to convince rational people of the superstitions we believe in, we should consider it "suchness" they are disappointed, unsatisfied, incredulous & even fall on the floor rolling in laughter. We should have equanimity rather than have the craving expecting we can convince others.

how would you answer this question so that it can enlighten/delight/pacify someone from a non-Buddhist tradition (with some inclinations towards an eternal soul).

If you develop samādhi (undistractible lucidity perfected with meditative training) to a high level, you can witness for yourself what happens to a soul at the moment of physical death.

If you don't (develop samādhi), then all you have is just hearsay, beliefs, and views.

Yogis with developed samādhi from all religions, Hinduism, Taoism, Essenes, christian mystics, etc., all witness rebirth of the "soul" as a real observable phenomena.

The difference with the Buddha, was he found that the "soul" is conditioned, impermanent. With the proper understandig and training, one can bring an end to the infinite cycles of rebirth and understand what a "soul" really is.

Answers to comments:

@Desmonica Sin, see anatta lakhana sutta for example. The 'atta' (self), or mama (mine making) are two of the terms the Buddha uses to describe the process of identifying with our mind/body complex, and that's the intersection of where other religions refer to 'soul' lucid24.org/sn/sn22/sn22-v01/index.html#s59

This answer sounds like Hinduism. – Dhamma Dhatu

@ Dhamma Dhatu - See the OP - they asked how would you answer someone from another religion, not what technical Buddhist terms you would use that they wouldn't understand anyway.

There is a two-fold answer to what is it that reincarnates. It is a two-fold answer because the answer is given via two-truths theory. Conventional truth and ultimate truth. (Here we are given that this is about Buddhism.)

Conventionally speaking, Buddhism does not accept soul as a thing which exists. But it accepts the psychological self. It is this person that practices dharma and achieves enlightenment. It also accepts that there are mental moments, and therefore a mental stream. It is this mental stream which arises in different bodies, depending on its karma. Cittamatra school labelled this as alaya-vijnana. It has often been translated as store-house consciousness (but please be cautious about the translation).

From the stand point of ultimate analysis, there is no self, there is no dhamma, there is no Buddha, there is no alaya-vijnana, there is no this, there is no that......therefore there is no thing to be reincarnated. In fact, from the standpoint of ultimate analysis, there is no reincarnation.

What your friend asked, and a lot of people ask, is (amongst other things) a confusion. There are things which exist conventionally like table, chairs,laptop, laws of nature, beings. And then there are things which do not exist even conventionally like unicorns, horned-rabbits, souls, etc. Reincanation process, which is taken to be valid conventionally, is explainable by things that are also held to be existing conventionally. It does not require a fictional thing like a soul.

how would you answer this question so that it can enlighten/delight/pacify someone from a non-Buddhist tradition

I would say, it is information that is reborn, and the "medium" that carries the information after the break-up of the body is the world itself. The imprints of previous existences are implicitly "stored" or "encoded" in the matrix of people, things, and activities. So there's some not-insignificant blending of information going on, as well as loss, so it's not like the same person reincarnates as-is - and yet there's enough continuity that eg Russians remain Russians and Indians remain Indians generation after generation, carrying their good and bad karma along.

As per the Questions of King Milinda:

--“Can there be any rebirth where there is no transmigration?”
--“Yes there can, just as a man can light one oil-lamp from another but nothing moves from one lamp to the other; or as a pupil can learn a verse by heart from a teacher but the verse does not transmigrate from teacher to pupil.”

...

The king said: ‘He who is born, Nāgasena, does he remain the same or become another?’

[Nagasena:] ‘Neither the same nor another.’

[King:] ‘Give me an illustration.’

[N] ‘Now what do you think, O king? You were once a baby, a tender thing, and small in size, lying flat on your back. Was that the same as you who are now grown up?’

[K] ‘No. That child was one, I am another.’

[N] ‘If you are not that child, it will follow that you have had neither mother nor father, no! nor teacher. You cannot have been taught either learning, or behaviour, or wisdom. What, great king! is the mother of the embryo in the first stage different from the mother of the embryo in the second stage, or the third, or the fourth ? Is the mother of the baby a different person from the mother of the grown-up man? Is the person who goes to school one, and the same when he has finished his schooling another? Is it one who commits a crime, another who is punished by having his hands or feet cut off ?’

[K] ‘Certainly not. But what would you, Sir, say to that? ’

The Elder replied: ‘Neither I am what is now the grown up, nor was I what was the tender tiny baby, flat on its back. But all these are tied in one by means of this body.’

[K] ‘Give me an illustration.’

[N] ‘Suppose a man, O king, were to light a lamp, would it burn the night through?’

[K] ‘Yes, it might do so.’

[N] ‘Now, is it the same flame that burns in the first watch of the night, and in the second?’

[K] ‘No.’

[N] ‘Or the same that burns in the second watch and in the third?’

[K] ‘No.’

[N] ‘Then is there one lamp in the first watch, and another in the second, and another in the third?’

[K] ‘No Sir. But thanks to that lamp the light shined all the night through.’

[Nagasena:] ‘Just so, O king, does the continuity of dharmas connect. One emerges, another dissolves, connecting as it were without [a clear boundary between] the previous and the next, thus the former-consciousness and the next-consciousness cannot be categorized as either the same nor as different.’

...

What is it, Nàgasena, that is reborn?”
“Namarupa.”
“Is it this very namarupa that is reborn?”
“No, it is not, but by this namarupa deeds are done and because of those deeds another namarupa is reborn; but that namarupa is not thereby released from the results of its previous deeds.

I would answer that it is the exact same thing - no more, no less - that passes from moment to moment in this very life and let them ponder what that is or is not. This would lead to further questions and ideas about what it is that passes from moment to moment. It isn’t a soul, but it is not nothing at all either. The same is true for what passes from life to life.

This answer will probably be disappointing for some and seen as evasive by others, but it gets at a deep and important point. What’s more, it turns the questioner to pondering anatta in this very life and what it means.

This answer does not give an affirmative to what it is that passes, but it rejects the two extremes and encourages the questioner to find the middle way.

PS: To illustrate the two extremes: supposing that there is something substantial and inherent that passes from moment in this very life is eternalism. Supposing that same substantial thing then disintegrates at the moment of death is nihilism. Supposing that there is nothing whatsoever that passes from moment to moment in this very life is nihilism and it contradicts conventional existence. Supposing that there is nothing whatsoever that passes from life to life is also nihilism and betrays that you believe that what passes from moment to moment in this very life is substantial that is then annihilated at death. The truth is found in the middle, but it is very hard to see.