to stay on: to remain in a place, or keep on in an action or condition, without stopping
At its root this is the ordinary verb for staying or remaining, μένω, with a prefix that pushes it toward staying on, keeping at it, or continuing further. The picture is always the same: someone does not move off or leave off, but holds where they are.
That one picture stretches across three everyday uses. It can mean staying put in a place for a stretch of time, as travelers do when they lodge somewhere for a few days. It can mean keeping on with an action already underway, like a crowd that goes on asking or a man who goes on knocking. And it can mean holding to a course or condition without giving it up, which is where the word gains its moral weight.
That third use is the one Paul reaches for again and again. A person may keep on in kindness, keep on in faith, or, just as easily, keep on in sin or in unbelief. The word itself is neutral: it simply names the refusal to stop. Whether that persistence is good or ruinous depends entirely on what a person is persisting in.