There are different ways of saying agree to disagree that influences how it might be perceived. It was first recorded in a letter of 1750 by George Whitfield, who stated
After all, those who will live in peace must agree to disagree in many things with their fellow-labourers, and not let little things part or disunite them."
It was used still earlier (1704) without the exact phrasing in a sermon by John Piggott:
And now why should we not agree to differ, without either enmity or scorn?"
To not use a well-intentioned, apt, and old phrase because some abuse it seems like throwing the baby out with the bathwater.
- We'll just have to agree to disagree, won't we? - agreed, insincere and unconceding.
- Let's agree to disagree. - more useful, polite, and still allows the other in the conversation to press their point, again, if they are obstinate.
- Perhaps we can agree to disagree? - I think this is a sincere attempt to bring unhelpful disagreements to a close in a friendly manner. Concedes that both arguments might have merit but one or both of the participants can't stop repeating themselves, thinking the other would agree if only they understood what was being said. The problem here is equating understanding with agreement, a very common error.
However, if you're looking to a more polite phrase to end an argument, I might propose
appeal to humor
Well, I've already said far more than I know about this!
appeal to a wiser person
I wish Confucius were here to tease out the truth. or We might need Solomon here; Lord knows I do. (This infers that the truth is around but speaker concedes he's not wise enough to find it.
appeal to relationship
You're a good (state relationship) and I respect your opinions. I don't want to (taint/tarnish/sully/stain/blot/mar/soil/muddy/damage/harm/hurt/besmirch) our (relationship example) over this.
appeal to peace
We are both peaceful people; let's give this up. or Let's move to more peaceful places.
appeal to better uses of your time
We're just not using our time to the best effect. Let's go knock back some beers at a good place I know.
appeal to honor (HT @Patrick M)
Let's settle this like nerds. I challenge you to a duel (with video games, board games or card games)!
Let's settle this like men. I challenge you to a (race, game of basketball, tennis, fighting match)!
c.) manly men
Let's settle this like real men. Whoever can (pick up the first chick, eat the biggest steak, bench press the most) wins the argument!
appeal to pride
I admit I just can't understand your point. I'm hopeless.
appeal to the ideal
In a perfect world, we would agree.
appeal to higher power
Let's let (God/other) solve this and have harmony until then.
appeal to authority
I wonder what (authority figure of choice) would say.
appeal to relative importance
The world won't stop if we disagree. This is not that important in the grand scheme of things.
Anyway, the possibilities are endless if you really only seek to end the argument. But you'll see that it may well cost you something to pursue peace, and it may not be as effective as just ending the argument by agreement. If you're willing to give something up, by all means, use that. But if you just want peace, I think "Please, let's agree to disagree" gives both parties an out without losing anything.
*Editted to add excellent suggestions by Patric M.