a boast: what a person can point to as a reason for pride, and the pride voiced in it
This noun belongs with the verb "to boast" (καυχάομαι), and Greek builds a noun of this shape to name what an action produces rather than the action itself. So it normally points at the thing a person boasts about, the ground under the boast, while the companion noun for the boasting itself is καύχησις. Paul does not police that line: he lets this word slide over into the boasting too, and swaps the two nouns inside a single argument.
What holds the range together is that a boast here is handled almost like property. It can be handed to someone else, drained until it is empty, or gripped firmly to the end. Because a boast and the thing it rests on are bound so tightly, one word covers either, and the question to ask of a verse is not just whether someone is boasting but what he is pointing at.
That is why the word is not automatically a fault. Paul tells one church that they are his boast and he is theirs, and hopes another church's pride in Christ will grow when he visits. To a third he says flatly that their boasting is not good, because what they were pointing at was their own tolerance of an ugly situation. The word takes its color from whatever stands behind it.