Is he talking about a do-over? You can't do that, that's not how the game is played...
Proper Chessiquette?

Being the perennial nice guy, I took the Q and offered a draw which he accepted. If it happens again, I will play it through. Have not rechallenged yet, but I don't like the site anyway, it's slowchess.com This site has a much nicer set up.

It might depend on where the queen was close to. If a very reasonable queen move was on the same diagonal, and it was touch move then he could have slipped it on a covered square.

Interesting choice, I like it. Take the queen and offer a draw. I bet him being higher-rated had something to do with it - you can come off looking like the nice guy (and you are a nice guy) and you still get something out of it.
There's some other interesting choices. How about these. Let's say that your opponent made something which was obviously a mouse-o or type-o, such as playing 1.e4 e5 2.d4 d6 3.dxe5 dxe5 4.Qd7+
- Opponent is a 200 point lower rated player in a blitz game.
- Opponent is a GM in an online blitz game.
- Opponent is equal rated but you just lost the previous game after having a won game.
- Opponent is equal rated but you just won the previous game after having a lost game.
- Opponent is about equal rated, you've been trading blitz wins and losses, and the position on the board is about equal.

Reminds me of a draw offer I made -- at a club blitz game I blundered badly and was going to get absolutely crushed by an attack on my king when he left his queen unprotected, so I paused and said "this game is awful, I offer you a draw" seeing what he'd done he accepted and we both left to get a drink, use the bathroom, etc -- upon returning we both had realized that he could have won anyway, we set it back and and sure enough his position was still good, so I swindled a draw in a very funny way.
Well, talking about the take-back vs. no take-back debate, I also think, personally, that one should take into account the nature of the game being played: is it an official rated/club/tournament game etc. or is it stakeless? To be honest, I don't think that requesting take backs is a terrible thing to do in an unrated game, as the primary purpose would be for learning purposes or just for one's own fun/sense of aesthetics. In such cases, where there are no stakes, I think that requesting a take back, is acceptable, as you can then explore where a potentially interesting or instructive game might develop and, if refused, you could either play on or resign, no hard feelings. In such cases, I don't think either side loses anything out of the situation.
Now, when the game is official, and stakes do come into the picture, I do agree that requesting takebacks is a fairly big transgression.

"I hung my Queen, and he's such a lousy player he didn't even see it."
Just as you're very careful when moving or exposing your Queen OTB, you should be just as careful online. And the mouseslip excuse, don't buy it. When you move such an important piece, you click the piece, and you then click the square you want it on. If you don't follow this simple precaution, too bad.
Chess is intellectual honesty - that means the game happens in your head. How would a blindfold player take back a queen he had announced to the wrong square ... ? He wouldn't.
And that twelve year old kid who began to cry when he dropped a Knight, mentioned just above?
"He, he. I pulled my "crying kid" routine to make him feel sorry for me, and he bought it."
There are no take-backs, it's a brutally honest game.

Being the perennial nice guy, I took the Q and offered a draw which he accepted. If it happens again, I will play it through. Have not rechallenged yet, but I don't like the site anyway, it's slowchess.com This site has a much nicer set up.
“You cannot play at Chess if you are kind-hearted”
(French Proverb)

i was playin' on chesscube, an' had a really dodgy old mouse which was forever doin' its own thing; on a simple an' obvious recapture it dropped me bit one square short, so I explained the situation to me oppo, who refused the take-back - leaving me a rook down. I got real angry an' checkmated the divot about twelve moves later an' the cheeky bastard had the temerity to ask me for another game !!! I told him "I only play with proper people" and whacked him straight onto me 'blocked' list
I took someone's queen with my pawn a few hours ago on live chess and he was in the driver's seat for that game at that point. He immediately cited mouseslip and asked for a draw. The game was ruined for me at that point as he kept going on about a screw up. It could have been a bad play just as easily as a mouseslip, in my opinion, who knows? I just moved the pawn the play before and he might have simply missed it. He said we should draw. I sent the draw request and immediately logged out. I just logged back on and it shows I lost the game by resignation? Isn't a draw a tie? I'm relatively new to livechess.
In hindsight, if this happens again with an opponent I would play on. A mistake's a mistake. I learned that lesson the hard way.

A draw is a tie, but if he didn't accept it before you logged out then the site would count it as a resignation.
It seems like mouse slips are an unfortunate part of the game in the online world. Its good that you learned your lesson, because you really never know. Just turn the chat off next time some fool gets mouthy.

In chess there is this "take-back" option in case of the site doesn't have the option in its software...
You agree with your opponent that you do a reverse-possible officer move, he take back his officer, and you yours, and the position is exactely as it was two moves ago, and him about to move. It doesn't work with all sorts of moves of course, and there is this slightly difference that draw by repetition is a tiny bit closer than before, or castling possibility could be destroyed.
And you have to trust the "gentleman" on the other side of the board.
Being the queen-dropper, I would resign. In my opinion I would not accept such an offer from my opponent, unless it was an unrated game. If on the other side of the board I would take the queen, since a game is a game, and a war is a war. And I would expect no mercy from him eighter... What you would like to explore, could be explored in a post-mortem anyway.
I'm on the side of those who say it was rude of him to put you in this position. Bad form on his part. He should be the one deciding what to do (play on or resign). There is no way around the 'whingey whiner' factor here.
Exactly. I think it's offensive that he even sent the email. What kind of weirdo, even. I don't mind an "oops" or "wow, I can't believe I did that" but regardless of the reason...I think it's pathetic when someone feels the need to even mention a mouseslip. I play touch move on here...and I've experienced the very same thing a few times. Once, even my electricity went while I was mid move and the piece ended up on an insane square...fed to my opponent. No, I did not feel compelled to write him about the bizarre incident or to ask for a draw...I rolled my eyes at my luck and carried on...as anyone with any decency would do. Or resign.
So tell us...what did you decide?