I highly doubt it! The child will definitely have to return to the country of origin to have their petition processed through the foreign consulate. They will be required to appear for a visa interview there.
I'm also a minor, and have tried applying for a SSN (because I need to do community service - high school requires it) but failed to get one. I also doubt it, but only because I've tried.
Maybe, if one of their parents or older siblings (older than 21) have a visa ("green card") or is a naturalized US citizen, they may be able to get the minor a visa. And I think that the minor will be able to stay here, and if not they can lie to immigration and say that the minor is in their home country until the visa is aproved.
It all depends, if the minor has parents who are residents or citizens they can petition for him/her on a I-130 form, if none of those chances are that he will have to get married to a resident ot citizen when hes older or so as we speak there is a bill in the house of represantatives called the dream act that can also help him out but its not a law yet.
** no. YOu need to discuss your options with an immigration lawyer. If you have family US citizens, if you came with a visa and overstayed, if you can apply for political asylum, if the migratory reform comes, if the dream act is aproved... all that issues are important you need to investigate by your self and get the right help: a lawyer.