Contrary to what the title may suggest, the subject of this post is not about the final word in a thought or sentence, but about getting "the last word" in an argument of some kind.
People seem to be hardwired to get the last word in. It's that final jab you get to take at another person. The last word always sticks with both parties. The person who got it really feels that they got to shove it in the other guy's face, and the other guy has to stew in the fact that the other guy got the last word in, even though there were so many possible comebacks.
When writing dialogue, you have to be careful with this. Speech is one of those situations where we always have to detract from reality in the name of good storytelling. Arguments get to be well-thought-out and perfect. People say just the right thing and the last person to speak wraps everything up (or produces whatever kind of feeling you intend).
I really recommend that dialogue be more on the idealized side in writing. I think of scenes where two people keep going back and forth to try to get the last word in and it just pisses me off. It's obnoxious and annoying. It's a scene where two stupid people take turns being stupid and your story gets put on hold while everybody is forced to watch the stupidity.
Dialogue is always best when it is short, sweet, and to the point. Decide who is going to get the last word and let them have it. There will be many more words to follow, so there just isn't that much to fight over.