Can someone atone for the sins of another, in the life after death, if they asked this grace from Our Lord before their own death in this life?

The Apostle Paul expressed such a desire in Romans 9:

9 I speak the truth in Christ—I am not lying, my conscience confirms it through the Holy Spirit— 2 I have great sorrow and unceasing anguish in my heart. 3 For I could wish that I myself were cursed and cut off from Christ for the sake of my people, those of my own race, 4 the people of Israel. Theirs is the adoption to sonship; theirs the divine glory, the covenants, the receiving of the law, the temple worship and the promises. 5 Theirs are the patriarchs, and from them is traced the human ancestry of the Messiah, who is God over all, forever praised! Amen.

In mentioning his desire, he does not expressly either affirm or deny the possibility of accepting extra suffering in order that others might be saved. However, he gives the impression that it is not possible, leaving all such decisions to God in His sovereignty. Voluntarily accepting suffering for people who are still alive is however beneficial:

15 But the Lord said to Ananias, “Go! This man is my chosen instrument to proclaim my name to the Gentiles and their kings and to the people of Israel. 16 I will show him how much he must suffer for my name.” (Acts 9:15-16)

At the heart of the problem is the suitability of the sacrifice. Christ is sinless, therefore his sacrifice was acceptable to God to atone for the sins of humanity. Anyone other than Christ offers a lesser sacrifice that is likely inadequate to the task.

Another problem has to do with the purpose of suffering and the fate of the elect. If suffering falls short of eternal damnation, then it is for a purpose of God's, to perfect a person and prepare them for heaven. The work of God must be completed before they can be admitted. No imperfect person will enter heaven. Only excess suffering not needed to accomplish this perfection could be removed. If there is excess suffering beyond what is needed, then that exposes God to the charge of being unloving.

If the suffering is that of eternal damnation, then you have the puzzle of election, a sticky topic. No elect person can enter Hell or God's soverignty is overthrown, thus a saved person cannot trade places and enter Hell on another's behalf. Likewise, if a person is not elect, nothing you can do will change that fact without also making God out to be a liar. If a person's destiny is not certain, not a part of God's sovereign plan, then there is no such thing as election, and the Bible expresses an untruth; Jesus is made out to be a liar.

Sources:

1. Sinning against an infinite God incurs an infinite cost. See https://www.catholic.org/encyclopedia/view.php?id=10849

2. St. Augustine of Hippo affirms a form of election:

Listen, thou ungrateful one, listen: “Ye have not chosen me, but I have chosen you.” Not that thou mayest say, I am chosen because I already believed. For if thou wert believing in Him, then hadst thou already chosen Him. But listen: “Ye have not chosen me.” Not that thou mayest say, Before I believed I was already doing good works, and therefore was I chosen. For what good work can be prior to faith, when the apostle says, “Whatsoever is not of faith is sin”? What, then, are we to say on hearing such words, “Ye have not chosen me,” but that we were evil, and were chosen in order that we might be good through the grace of Him who chose us? For it is not by grace, if merit preceded: but it is of grace: and therefore that grace did not find, but effected the merit.

More here:

https://reformedreader.wordpress.com/2014/08/18/augustine-on-unconditional-election/

Quoted from:

Augustine of Hippo. “Lectures or Tractates on the Gospel According to St. John.” St. Augustin: Homilies on the Gospel of John, Homilies on the First Epistle of John, Soliloquies. Ed. Philip Schaff. Trans. John Gibb and James Innes. Vol. 7. New York: Christian Literature Company, 1888. 353.

Can someone atone for the sins of another, in the life after death, if they asked this grace from Our Lord before their own death in this life?

The answer is No.

To elaborate, can that mother pray like this: " Lord, let me be given more time in purgatory , but spare my son of your divine punishment and let him be in heaven on the day of his death "

The short answer is No again.

Explanation

The soul after death has no "body" and the soul undergoing purification in purgatory has no "body" to offer as a sacrifice for atonement. A good read as foundational teaching is St.John Paul II Theology of the Body,to know the reason why God had given us a body animated by a soul.

According to author Christopher West, the central thesis of John Paul's Theology of the Body is that "the body, and it alone, is capable of making visible what is invisible: the spiritual and the divine. It was created to transfer into the visible reality of the world, the mystery hidden since time immemorial in God, and thus to be a sign of it."[1] A rational soul was given a body to merit graces while still alive and after death the soul has no longer the means to acquire graces.https://en.wikipedia.org/wiki/Theology_of_the_Body

Jesus the Logos was incarnated precisely to atone for our sins. Jesus Christ by His sufferings, passion, crucifixion and death had won for us infinite graces in atonement for the sins of all humanity. Jesus paid our ransom with His most precious blood up to last drop. Jesus needed a "body" as a means for atonement.

Atonement (also atoning, to atone) is the concept of a person taking action to correct previous wrongdoing on their part, either through direct action to undo the consequences of that act, equivalent action to do good for others, or some other expression of feelings of remorse. From the Middle English attone or atoon (“agreed”, literally “at one”), now meaning to be "at one", in harmony, with someone.[1] Atonement "is closely associated to forgiveness, reconciliation, sorrow, remorse, repentance, reparation, and guilt".[2] It can be seen as a necessary step on a path to redemption.[3]

Atonement in Christianity, in western Christian theology, describes beliefs that human beings can be reconciled to God through Christ's sacrificial suffering and death.[6] Atonement refers to the forgiving or pardoning of sin in general and original sin in particular through the suffering, death and resurrection of Jesus,[7][8] https://en.wikipedia.org/wiki/Atonement

Another word related to your question is "reparation", since another person will be offering a sacrifice for the benefit of another person. And reparation can only be accomplish by a soul using his body as a means of sacrifice.

A good read will be St.Louis De Montfort writing on reparation;

Reparation is an ambiguous term. Even in profane use it may be employed in the sense of repair of a damaged object or an act of justice whereby payment of some sort is made for damage done. In religious use it has a variety of meanings. It means principally the work of redemption accomplished by Our Lord Jesus Christ, in the sense of "repairing the damage done" by Adam’s revolt and the sin of his progeny; Christ restores us to God’s friendship. The term is also used in a generic way for restitution for injuries, usually when moral theology cannot measure precisely what such payment would entail. In popular devotional literature and also in ascetical theology, reparation is the making of amends for insults given to God through sin, either one’s own or another’s. Through Saint Margaret Mary’s devotion to the Sacred Heart of Jesus—a devotion stemming through Saint John Eudes back through Saints Gertrude and Mechtilde and Saint Francis’ devotion to the Five Wounds and Passion of Christ—"reparation" took on a more distinctive meaning. Saint Margaret Mary saw Christ’s heart and his love ignored and ridiculed; the response of man is to be reparation through adoration, prayer, and sacrifice. https://www.ewtn.com/library/montfort/Handbook/Reparat.htm

To console your good motive on the question, the soul in purgatory can pray for us and plead for God to help us in our needs. There are a lot of stories where a soul in purgatory has an interaction with the living like the story of Maria Simma who was visited by a number of souls in purgatory. Also, Susan Bertone books in purgatory is a good reference too.

If you are familiar with the prayer for the poor souls in purgatory like the "Read me or Rue it" and also the famous Saint Gertrude. In God mysterious ways the poor souls in purgatory have been given a grace to know who are the people who are offering prayers & sacrifices for them and in turn these poor souls out of gratitude also in God mysterious ways had been able to help us in their pleading.

And the Catholic Church teaches that the souls in purgatory after purification enter in heaven like the saints. As St.Paul teaches that all of us will receive a crown in heaven and will be like Christ, our purified souls will be glorified.These poor souls now glorified in Heaven becomes for us generous intercessor. We belong to a Catholic Church who are composed of Church Triumphant, Church Sufferings and Church Militant and all this church are inter-connected in the beautiful Wisdom of God.

In closing, the poor souls can pray for us in a form of pleading to God but they can no longer make atonement for our sins. The poor souls pleading in behalf of us is different if a poor souls would desire to atone for our sins.