OK, well you did offer. I agree with ECB it's awkward but not as crazy as you make it seem given your offer to replace the lamp (which would be presumably more than $20). I think you put yourself in this situation by blaming your kid when you really didn't know who did it. If you are going to assume responsibility then don't be surprised when someone takes you up on it.
To be clear, I still think it was weird for your friend to take money at all (nevermind the request for an arbitrary $ amount). I think most in her sitch would realize that's a risk you take when you leave 10 kids alone in a room. Lol
I was just about to type something like this, but thankfully @booby had it covered. As she said, fault is irrelevant at this point since you accepted blame on your kid's behalf.
As for the money, personally, I would not take money from a friend over an accident. Accepting the risk of damage is part of hosting. BUT, asking for $20 is not any ruder than just accepting the gift card you offered. To me, it actually makes sense. The lamp isn't totally destroyed, but it's value has been diminished, so she is accepting that cost from you. There are tons of lamps at Target in the $50-$100 range, so I wouldn't assume she's making money from this. And maybe she needs her H to tell her how much he spent on the lamp initially?
So, yeah, I sort of do think you're wrong to be annoyed by this. It would be different if you hadn't offered, but once you opened that door, this really wasn't an odd outcome.
I'm sure Julez was sincere in her offer but as some one else upthread said, any decent person would take that as pro forma and brush it off.
Of course, and if she was asking if she should accept someone else's offer, we would say that. But you can't offer and then be (justifiably) annoyed when they accept!