Buddha teaches the extinguishing of suffering, what if an individual is okay with suffering?

What does buddhism say about someone, after learning about suffering, chooses to accept it?

It's difficult for the average typical worldling to "choose to accept" suffering. The 3 poisons come pre-packaged with the human operating system. So, when facing the unpleasant, what's the typical reaction? Resentment, anger, or frustration. Facing the pleasant? Craving, clinging, lusting after. So, interestingly, the only reliable way to be able to choose to accept suffering is... the cultivation of the Buddha Dhamma through Sila/Samadhi/Panna so that gradually one's loving-kindness, compassion, joy, AND EQUANIMITY are strong enough to enable one to do so. Without the Path, "choosing to accept suffering" is just another coping mechanism to mask the issue while all the underlying unwholesome tendencies for lust, anger, ignorance remain simmering below the surface and will strike again when new conditions arise.

Good householder,

"Whatever is felt comes under stress (dukkha)."

Alone

possible never investigated but actually what ever beings are doing, day in, day out, in every act by body, speech and mind, is to aviod and escape stress&suffering (dukkha). Even if there is one seeking suffering, actually it's for the sake of well-being.

Hence the idea of telling oneself that it's fine as it is, is also nothing else then another try to escape it.

That is why the very root cause of suffering isn't will (to not escape it) but not-knowing, delusion, ignor-ance (in regard of suffering and the way out).

Whoever sees
        pleasure as stress,
sees    pain as an arrow,
sees    peaceful neither-pleasure- 
nor-pain
        as inconstant:
    he is a monk
    who's seen rightly.
    From that he is there set free.
        A master of direct knowing,
            at peace,
        he is a sage
        gone beyond bonds.

(iti)

[Note that this isn't given for stacks, exchange, other worls-binding trades but for an escape from suffering by it]

In Buddhism, the only reason why someone embraces suffering / unsatisfactoriness and chooses to co-exist with suffering / unsatisfactoriness, is due to clinging or attachment to sensual pleasures and existence (the idea to become something or somebody).

From SN 6.1:

“This principle I have discovered is deep, hard to see, hard to understand, peaceful, sublime, beyond the scope of reason, subtle, comprehensible to the astute. But people like attachment, they love it and enjoy it. It’s hard for them to see this thing; that is, specific conditionality, dependent origination. It’s also hard for them to see this thing; that is, the stilling of all activities, the letting go of all attachments, the ending of craving, fading away, cessation, extinguishment. And if I were to teach this principle, others might not understand me, which would be wearying and troublesome for me.”

This is further elaborated in MN 75:

"Now suppose that there was a leper covered with sores & infections, devoured by worms, picking the scabs off the openings of his wounds with his nails, cauterizing his body over a pit of glowing embers. The more he cauterized his body over the pit of glowing embers, the more disgusting, foul-smelling, & putrid the openings of his wounds would become, and yet he would feel a modicum of enjoyment & satisfaction because of the itchiness of his wounds. In the same way, beings not free from passion for sensual pleasures — devoured by sensual craving, burning with sensual fever — indulge in sensual pleasures. The more they indulge in sensual pleasures, the more their sensual craving increases and the more they burn with sensual fever, and yet they feel a modicum of enjoyment & satisfaction dependent on the five strings of sensuality.

Also from the Dhammapada:

273. Of all the paths the Eightfold Path is the best; of all the truths the Four Noble Truths are the best; of all things passionlessness is the best: of men the Seeing One (the Buddha) is the best.

274. This is the only path; there is none other for the purification of insight. Tread this path, and you will bewilder Mara.

275. Walking upon this path you will make an end of suffering. Having discovered how to pull out the thorn of lust, I make known the path.

276. You yourselves must strive; the Buddhas only point the way. Those meditative ones who tread the path are released from the bonds of Mara.

And also:

335. Whoever is overcome by this wretched and sticky craving, his sorrows grow like grass after the rains.

336. But whoever overcomes this wretched craving, so difficult to overcome, from him sorrows fall away like water from a lotus leaf.

The word used by the Buddha is "dukkha", not suffering.

Dukkha is (by definition) the pain you experience when things are not how you want them to be, either objectively wrong or subjectively wrong, and you can do nothing to fix it.

For someone you describe, suffering is not wrong, therefore for them it is not dukkha. But any other aspects of life having a character of conflict between "is" and "should" will still be experienced as pain by this person. So liberation from that is still relevant for them.

Of course if they have absolutely no trace of conflict in their mind, then for them there's no dukkha, but what can we say about their environment? Do they have the insight to avoid creating pain in other people? They themselves may see suffering as peace, but does that hold for the others in their life, and if not, what does it tell about them?

Coming back to your question, this relationship between suffering and suchness is one of the key topics of Mahayana. Indeed, if suffering is an inevitable part of the relative life, and we accept the relative in its entirety as a manifestation of the absolute, doesn't it mean we accept suffering, too? And what happens to the painful aspect of suffering when we accept it? Hence the Mahayana's motto, "Samsara is Nirvana".

In short, your question is a valid one and the answer is in Mahayana's concept of the unity of the two truths, the relative and the absolute.