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.
