Summary: Paul's whole plea to Philemon is to act the way the loving Father acted: to welcome home the beloved son; to restore him to fellowship with the family. There will be, and have been, occasions when we'll be called to respond to wrongdoers the same way.
There's something quite different about reading this letter compared with all Paul's other letters. In all the others, even those addressed to individuals, the focus is on the Church where they're ministering. But here there's a much more personal feel to the correspondence. You can feel the warmth of Paul's feelings towards Philemon and probably his wife and son, Apphia and Archippus. Paul hadn't visited Colossae but it's likely that Philemon was in Ephesus at the same time as Paul and was converted there. It certainly sounds like Paul had spent time with them and he clearly wishes he could be with them enjoying their hospitality rather than being in Rome experiencing the emperor's.
But there's a much more serious reason for writing than booking the guest room. While he's been in Rome Paul has met up with a runaway slave named Onesimus. Onesimus has become a Christian under Paul's teaching and is beginning to turn his life around. In fact he's become one of Paul's carers, probably bringing him food and other supplies while he's in prison in Rome. But his presence with Paul raises an ethical dilemma. What should Paul do? Should he hand him over to the authorities as an escaped slave or should he keep quiet and just enjoy the support he provides.
Interestingly he doesn't raise the theological question of the rights and wrongs of slavery. This is one of those difficulties that face us when we read the New Testament with 2000 years of history behind us. Does Paul think slavery is OK or is there another reason for ignoring the question? We'll find the same question arising in a couple of months' time when we look at Ephesians 5 (quickview)  where Paul addresses husbands and wives within the context of a highly patriarchal culture. There, as here, he doesn't directly critique the cultural or societal norm, but he does lay the groundwork for those norms being reconsidered at a later date.
Here, in the context of an economy that's entirely dependent on slavery he chooses not to raise the question of whether it's right or wrong. In Galatians 3 (quickview)  he'll say quite clearly that there's no distinction as far as the gospel's concerned between slave or free but here he prefers to address the question by looking at the relationships involved.
He speaks with great care and tact, doesn't he? He begins by establishing his love and respect for Philemon. What he's about to suggest is a big ask. Not only did the economy rely entirely on slavery but the management of slaves required masters to be ruthless in dealing with any slaves who ran away. Without Paul's help Onesimus may well have been put to death or at least would have been severely flogged. But Paul wants the exact opposite outcome. So he speaks carefully. He reminds Philemon of their relationship. He's like a brother to him; his wife is like a sister.
And he prays that the sharing of his faith might become effective when he sees all the good that they might do for Christ (v6).
Notice that the crucial characteristic here is love (v7). He mentions love 2 or 3 times. You don't normally associate love with being mistreated do you? - unless there's some other psychological issue going on within a relationship of course. So if Philemon will treat Onesimus as someone he loves it'll make all the difference, won't it?