Considering that children of this age are allowed to enter at all (a separate issue) I saw only an act of kindness when she was given a chance to sing again. This was not the tantrum of a spoilt little madam, just the very normal reaction of a little girl who made a mistake, faltered, and became distressed and overwhelmed.
Some comments on this thread have been unbelievably harsh on her, I think.
In the circumstances, I think the least worst way forward was to let her start again - I think it was likely a kindly and human impulse from the judges, I'm sure they would much have preferred she vanished never to be seen again (especially since they must have realised the probability was high of her finishing in vote-off position and making it all awkward, as indeed she did.)
I didn't mind them giving her the place in the end, given the choice - the singer in the other runners'-up spot was dire tonight - in fact it's been a strikingly talentless year on the whole, with the obvious exceptions - but I think if I were her mother I might gently suggest she finish on the high note of last night's applause and cheers and kind words and pull her out of the final, with a face-saving 'you can have another go next year if you want to, when your voice is even stronger' or similar. Relief all round if she did (don't suppose she will, though.)