Is there a specific word in Buddhism that refers to a "second chance" or perhaps redemption?
Skill in offenses and skill in rehabilitation from offenses.
Āpattikusalatā ca āpattivuṭṭhānakusalatā ca dn33
I think this is close to redemption from wrongness. As i understand it, the semantics are close to 'rising up & emerging' as a recovery from wrong-doing.
Perhaps it's words including nirodha (cessation, disappearance, extinction) ...
Now this is the noble truth of the cessation of suffering.
Idaṁ kho pana, bhikkhave, dukkhanirodhaṁ ariyasaccaṁ
... and vimutti (freedom, release, deliverance) ...
The ocean has just one taste, the taste of salt.
Seyyathāpi, bhikkhave, mahāsamuddo ekaraso loṇaraso;
In the same way, this teaching and training has one taste, the taste of freedom.
evamevaṁ kho, bhikkhave, ayaṁ dhammavinayo ekaraso vimuttiraso.
I think that the word "redemption" is too explicitly Christian. Its ordinary meaning is "gaining possession of something in exchange for payment, or clearing a debt", and theologically it refers to the Christ's "paying for people's sins in order to liberate them from hell", which I think has no direct equivalent in Buddhism.
A Buddhist equivalent is simply cessation -- which, I understand as including "cessation" of being harmful, so being and becoming harmless, and cessation of remorse, and so on.
One example is in the Angulimala Sutta (MN 86) -- maybe not a relatable example because it's an extreme, but it shows that even in this extreme case what's required is cessation:
... There is a bandit in my realm, lord, named Angulimala: brutal, bloody-handed, devoted to killing & slaying, showing no mercy to living beings. He has turned villages into non-villages, towns into non-towns, settled countryside into unsettled countryside. Having repeatedly killed human beings, he wears a garland made of fingers. I am going to stamp him out."
"Great king, suppose you were to see Angulimala with his hair & beard shaved off, wearing the ochre robe, having gone forth from the home life into homelessness, refraining from killing living beings, refraining from taking what is not given, refraining from telling lies, living the holy life on one meal a day, virtuous & of fine character: what would you do to him?"
"We would bow down to him, lord, or rise up to greet him, or offer him a seat, or offer him robes, almsfood, lodgings, or medicinal requisites for curing illness; or we would arrange a lawful guard, protection, & defense. But how could there be such virtue & restraint in an unvirtuous, evil character?"
Now at that time Ven. Angulimala was sitting not far from the Blessed One. ...