favor freely shown by someone who is free to give it, and that favor made real in a gift, a standing, or a power at work in the one who receives it
This is the heart of the word and where most of its New Testament uses fall. At its core it names a favorable disposition: one person, usually the greater or wealthier, looks kindly on another and acts for their good. In the everyday world of the first century this was the language of the public benefactor and the king who showered cities and people with gifts, so the word never floated free of real relationships. It is the word for finding favor before God or a ruler, and the word for the generosity that gift expresses.
Paul takes this ordinary social favor and makes it the name for the whole saving act of God in Christ: God gives the supreme gift to people who have done nothing to deserve it. That single move, giving to the unworthy, is what marks his use off from the usual giving to fitting and honorable recipients. The favor is so central it becomes the very sphere a believer lives in, a standing one is brought into and stands in.
From there the word stretches naturally. The same favor takes concrete shape as a gift, including the relief money Paul gathered for poor believers in Jerusalem. It shows up as a power at work, equipping a person for service beyond their own strength. And it opens and closes nearly every letter as a blessing on the readers. These are not different words but one favor seen from different angles: the disposition, the gift it produces, and the effect it has on a life.
The ancient world heard all of this inside a circle of giving, receiving, and returning. Paul keeps the circle: the gift comes with no prior conditions of worth, yet it is meant to reshape the receiver into someone who lives a changed, loyal life. It is, as it has been put, a gift given without conditions but not without expectations.