to give back in return: to settle an account by handing back what someone's own action has earned, whether a kindness answered or a wrong punished.
The word is stacked out of three pieces that all push the same way: give, give back, give back in return. What it names is the answering half of a transaction. Something was put into your hands, a favor or an injury, and this verb is the act that evens the books.
That is why one Greek verb covers what English usually splits between reward and punishment. The logic is identical in both directions: what comes back matches what went out. Paul can use it of God paying people back at the resurrection of the righteous, and then of God paying back the people who are persecuting the church, without reaching for a different word.
In the New Testament the one doing the paying back is almost always God. People turn up mostly as unable to do it. The poor guests in Luke 14 have nothing to hand back, and Romans 11:35 asks who could ever have given God something first, so as to put him in their debt. That is the whole force of the line quoted twice, "Vengeance is mine, I will repay": settling accounts is deliberately taken out of human hands and left with God.
One use runs the other way. Writing to Thessalonica, Paul asks what thanks he could possibly give back to God for these people, treating gratitude as the only repayment a gift like that can receive. The plainer verb for paying what is owed (ἀποδίδωμι) is the everyday word for wages, vows, and debts; this one keeps the sense that the payment is an answer to something already done.