What is (re)born? Did the Buddha explicitly state that it is "the mental idea of the self" that is reborn?

I don't know Yeshe (: But I think you don't exist as an identity, if you are always changing as a ''person'' and in body then you don't exist or you wouldn't change. What you really are is beneath what you believe and think. And because you aren't an identity that doesn't accompany you. Only your karma which is your pure essence I think.

Is like you are a buddha but as you may not still see it you just go trough life as actions direct you trough. And Karma.

From Snp 4.14 (translated by Ven. Thanissaro):

"I ask the kinsman of the Sun, the great seer, about seclusion & the state of peace. Seeing in what way is a monk unbound, clinging to nothing in the world?".
"He should put an entire stop to the root of objectification-classifications: 'I am the thinker.'

From the translator's commentary:

On objectification-classifications and their role in leading to conflict, see Sn 4.11 and the introduction to MN 18. The perception, "I am the thinker" lies at the root of these classifications in that it reads into the immediate present a set of distinctions — I/not-I; being/not-being; thinker/thought; identity/non-identity — that then can proliferate into mental and physical conflict. The conceit inherent in this perception thus forms a fetter on the mind. To become unbound, one must learn to examine these distinctions — which we all take for granted — to see that they are simply assumptions that are not inherent in experience, and that we would be better off to be able to drop them.

I agree with the answer by Andrei.

Whatever ignorance is left in the mind continues to arise. So not only the self idea but all the ignorance not removed by following the right path will arise.

That said, I would like to add that these higher teachings could be risky if parroted to everyone without regard to the person's stage of development. Some people who don't have a well-developed virtue and a thick attachment to self-love, like myself, could fall into confusion and/or abandon virtuous acts. So, if anyone is tied in that loop I urge you to read this question that I asked a while back; there are some good answers given there for the benefit of the Dhamma here and now.

I also think that is the reason why in some of the teachings the idea of Kamma is presented crudely as quoted below, to sound as if you are reborn of your own actions. It may sound wrong, but saying you are reborn, and that another 'mind and matter' are reborn, is the same, because you are a mind and matter with the same ignorance that gets reborn. So it is all semantic.

"'I am the owner of my actions (Kamma), heir to my actions, born of my actions, related through my actions, and have my actions as my arbitrator. Whatever I do, for good or for evil, to that will I fall heir'..

Generally, the suttas say "a being" is "reborn", as follows:

When the mind was thus concentrated, purified, bright, unblemished, rid of defilement, pliant, malleable, steady, & attained to imperturbability, I directed it to the knowledge of the passing away & reappearance of beings. I saw — by means of the divine eye, purified & surpassing the human — beings passing away & re-appearing, and I discerned how they are inferior & superior, beautiful & ugly, fortunate & unfortunate in accordance with their kamma: 'These beings — who were endowed with bad conduct of body, speech & mind, who reviled noble ones, held wrong views and undertook actions under the influence of wrong views — with the break-up of the body, after death, have re-appeared in the plane of deprivation, the bad destination, the lower realms, in hell. But these beings — who were endowed with good conduct of body, speech, & mind, who did not revile noble ones, who held right views and undertook actions under the influence of right views — with the break-up of the body, after death, have re-appeared in the good destinations, in the heavenly world.' Thus — by means of the divine eye, purified & surpassing the human — I saw beings passing away & re-appearing, and I discerned how they are inferior & superior, beautiful & ugly, fortunate & unfortunate in accordance with their kamma.

MN 4

The suttas say "a being" is merely a "view" of "attachment":

Why now do you assume 'a being'? Mara, have you grasped a view? This is a heap of sheer constructions: Here no being is found.

SN 5.10

'A being,' lord. 'A being,' it's said. To what extent is one said to be 'a being'?"

"Any desire, passion, delight, or craving for form, Radha: when one is caught up there, tied up there, one is said to be 'a being.'

"Any desire, passion, delight, or craving for feeling... perception... fabrications...

"Any desire, passion, delight, or craving for consciousness, Radha: when one is caught up there, tied up there, one is said to be 'a being.'

"Just as when boys or girls are playing with little sand castles: as long as they are not free from passion, desire, love, thirst, fever, & craving for those little sand castles, that's how long they have fun with those sand castles, enjoy them, treasure them, feel possessive of them.

SN 23.2

From Milindapanha:

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

Just like an adult "inherits" results of the choices made by the child - even though most of the original body cells are gone, similarly future existence "inherits" karma from the past existence.

--“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.”

It is information and causation that is reborn, or is re-embodied. Or, as Chogyam Trungpa said, it is our neuroses that are reborn.