What would the procedure be if someone had overstayed their visa here (meaning they lived illegally in the US), but they had a child here. Even thought they have overstayed and work illegally, what happens to the kid? Or what happens to the parents? The child is a of course a US citizen because of being born in the US, but what are the choices for the parents?

In a case like that- a good Immigration Attorney would be the best course of action.

Nothing changes for the parents' status because of the birth of a USC child.

Except that in 21 years, the child can petition for her parents.

The parents don't get any special immigration benefit as a result of having a US Citizen child. Under current law, they may wait until the child turns 21, and then the child may file a petition for them. At that time, they'd have to meet admissibility requirements. An overstay would generally subject them to a three or ten year bar, depending on how long the overstay was. I don't have a crystal ball, but my suspicion is that immigration laws will change sometime in the next 21 years, so who knows what the actual procedure might be by then. It's possible there will be no avenue for the child to petition for the parents at that time.

I don't know how Congress might change immigration law, but I suspect they will continue the present concept of avoiding incentives for people to have babies in order to gain immigration status. There would be political uproar and backlash if they changed the law to say, "Giving birth to a US Citizen child automatically gives anyone a legal path to citizenship".

As for the child, he's a USC because he was born here, and that status is irrevocable (14th Amendment to the US Constitution). The child might have also inherited the citizenship of his/her parents, depending on the laws of the parents' country(ies).

The parents may be deported if caught. There have been recent publicized cases similar to this. When the parents are deported, they may take the children with them back to their home country if the children have the right to go to that country according to the home country's laws. Or the children may be placed into foster care in the US. Either way, it's not a pretty situation, especially if the children don't speak the language of their parent's country.

In practice the Mothers are usually pretty safe, you will go a very long way to find an example of a Mother who has a USC Child being deported.

In fact there was a local story a year or so ago of a woman who was trying to surrender to ICE to go before an IJ. ICE did not want to know.

If it made any difference that they had a child in the US, most of the illegal aliens in the US would already be legal.

If a person entered the country without inspection and had more than 180 days of illegal presence (this is not overstaying a visa if there was no visa to begin with) a relative, fiance or spouse has to petition for them. Once the petition is approved (illegal presence is not a bar to petition approval) they would need to return to their country, interview, be denied at the interview for the illegal presence and illegal entry, and submit a 601 waiver with a letter proving extreme hardship to the qualifying relative.

If the person entered on a legal visa and overstayed more than 180 days, a qualifying fiance or spouse could petition for them and they could adjust their status in the US.

That's how the law stands today. If immigration reform passes, this could change.

Ooh, they did one here in Oregon a ways back--she was still nursing and they bounced her back. Much uproar and hand wringing and she was amazingly back inside of two weeks.
Married to a USC tho, not just on the kid's back.

That may have been true in the past, but operation "Return to Sender" has been active in the past month or two. They've definitely been deporting mothers of US Citizens recently. And under current law, they have the legal right and obligation to do so.

See this AP Story, for example, or this one, or here, or here.

ICE spokesman Marc Raimondi said all immigrants arrested by ICE are interviewed to determine if they are the sole parent of their children. ICE then can grant humanitarian releases

From your first link, I do not have time to look at the others.

I did not say it never happened, just so rare that when it does happen its news.