Minor piece is the correct answer.
Winning a minor piece vs winning an exchange? Which do you prefer?

Are you serious?
Like asking is it better to kiss a beautiful stranger or your sister...er...sibling.
*Edit
On second thought, is winning a queen for a bishop, knight, or rook considered winning an exchange?
Definitely the minor piece. Below 2K level I think it can sometimes be surprisingly difficult to convert an exchange into a win if your opponent has some compensation. On the other hand a free piece is pretty trivial.

On second thought, is winning a queen for a bishop, knight, or rook considered winning an exchange?
No, that's usually called "winning the game".

I'd rather have a checkmate, thank you. In case there's none, just show me the position- else it is another pointless comparison.

Pfren, I disagree. Players need to know the relative value of the pieces. All other things being equal, a piece is worth about three pawns, an exchange is worth about 1.5 pawns. (Some would say 2 pawns, but practice shows that's a bit optimistic.)
That's not pointless, that's basic chess knowledge.

I would take the piece, most of the time. Think like this. Both you and your opponent have 2 rooks and a bishop, but you can win material. Do you prefer 2 Rooks v Rook and Bishop or 2 Rooks and Bishop v 2 Rooks?
Normally you would take the second choice.

SmyslovFan:
Err... don't you think that someone with chess.com rating of 1600 should be able to count that deeply already?

On this site?
His blitz rating is 782.
Perhaps the person's trolling, but it's a legit question that beginners ask all the time.

re: op question, a comparatively advanced idea is that the power of capturing is equal among all pieces, only the mobility is different. So 2 vs 1 (or in this case, 1 vs none) is a consideration of its own.
For example a rook and knight vs a queen. The rook and knight can attack/defend any square twice while the queen can only do so once. In some situations (down material, but up in the number of pieces) this determines which side is better.
A bit of a tangent, but since this has already been answered thought I'd throw that out there.
This is wrong. Q vs B+N is almost always better. (There are some positions where the minor pieces can set up a blockade.) The Q is so powerful that it can even combat two rooks if the rooks aren't able to coordinate.

My god, can people read? I said "generally speaking". No kidding - it depends on position. That's why I thought I clarified. Okay, for the dense among us, if the position is starting out and you can choose the handicap. You can be a minor piece up or swap a knight/bishop for a rook.
Happy? Damn.

I agree when there are so many pawns in the beginning the rooks are hardly useful except for castling, if someone played a closed position they would defintly be at an advantage with a knight and bishop for a rook.

Generally speaking A > B unless B > A. In all other cases, A = B.
There is no such thing as "generally speaking" in chess, son. Piece values are relative to the specific position, and there is no such thing as a "generic position".
Here for example, winning a piece in an endgame with few pawns may lead to a theoretically drawn piece up ending, while winning the exhange may well win as the rook may gobble more material later. I could make an example, but it's really pointless trying to prove the self-evident.
You have the option to win a minor piece, yet you also have the option to win an exchange. Generally speaking, which would you prefer in most instances?