Paul is asking for God's mercy on Onesiphorus's family. And he is speaking of O in the past tense. It is obvious, in context, that O. is dead.
All of the times Paul uses the phrase "on that day" (someone receiving God's grace or some sort of reward") Paul means eternal life!
From all times and all places, for 1500 years and continuing all 2000+ years (excepting for the relatively small faction of the continuing-to-fracture-Protestant movement,) every Christian believed that that term meant being received into heaven, the ultimate reward.
Only when the Protestant movement started teaching that all truth must be found only in the pages of the Bible, did this term begin to mean some sort of reward given to us once we died and immediately went to heaven (because of a prayer we once said, not based on the state of our soul's charity with God and man at the time of their death. Notice I did not say based on all their "works" piled up.)
They dismissed 1500 years of Catholic belief in many truths given by Jesus orally to the Apostles (the faith once delivered to the saints.) Which continues to this day in the Magisterium (where we can look to for authority. Where do Protestants get their authority?)
The Bible itself (which is where Protestants claim all truth comes from) was given to all Christians by The Church that Jesus instituted on earth: The Catholic Church. It was never spoken about by Jesus at all. He commissioned his Apostles to go and tell everything Jesus had told them. He never said to "write this all down."
Why can you believe that the men who decided which books were in the canon (around 400 A.D.,) but do not believe what they themselves lived and believed? And changed the world with their lives and beliefs.
And they believed that Paul was asking for God's mercy on the soul of a dear, dead friend (No one can know for sure if someone is in Heaven, Hell, or Purgatory, so we ask for God's mercy on their soul in any case.)
The huge problem with the idea that Paul is only speaking of a reward (not eternal life) as something we will receive because of something we did above and beyond what is necessary, and not the reward of Heaven because we "fought the good fight" and "ran the race set before us" and persevered to the end, is that it makes living a moral life in charity with god and neighbor... optional!
If I can go to heaven and be with God forever and never have to change my behavior, with God's grace, through Jesus Christ, consistently changing the way I act, after my passions react to a situation, which takes a lot of thought, prayer and change.
Why would I bother?
God will always just forgive me, and actually, already has! I say "the believer's prayer" and all my sins are forgiven past, present, and future. Why do we wonder that Christians have a bad rep for being hypocrites? They never have to do a thing to change and their salvation is guaranteed!
Plus, praying for those who have gone on before us gives us the sense of community that God created us for. Not with those just living around us now, but "the great cloud of witnesses" who have lived on this earth before us and are now in Heaven (or purgatory) praying for us as well.
And saying that Paul is asking for God's grace on O. because he has fallen away, is not supported in the text at all. Nowhere does it say that O. fell away.
Conclusion: The Authority of The Church tells us that O. is dead and Paul is praying for a man no longer living on this earth, but alive in Heaven/Purgatory.