Child who entered under ESTA/VWP accompanying US citizen mother

First, as AndrewLazarus pointed out, there is no such thing as "eligible for US citizenship". Most likely he is already a US citizen, and has been since birth. The condition for whether he was a US citizen at birth are: 1) if he was born in wedlock to two US-citizen parents, that either parent had a "residence" in the US at some point in their life before the child's birth; 2) if he was born in wedlock to one US citizen parent and one alien parent, that the US citizen parent had been physically present in the US, any time in her life before the child's birth, for a cumulative total of 5 years, including 2 years after she turned 14; 3) if he was born out of wedlock, that the US citizen mother had been physically present for some continuous period of 1 year some time in her life before the child's birth.

In fact, a US passport can only be applied for someone who is already a US citizen, so if you are applying for a US passport for him, you must think that he is already a US citizen. Otherwise, he cannot just directly apply for a US passport. (If he weren't a US citizen, the ways to get him US citizenship are either to 1) petition him to immigrate to the US, i.e. become a US permanent resident, and when he becomes a permanent resident he would also immediately become a US citizen; or 2) go through the INA 322 process for naturalization for a child residing abroad, but that requires filing N-600K and getting that approved, before entering the US as a nonimmigrant and taking the oath while still in status; if that process hadn't been started, there wouldn't be enough time to complete it while in status on this trip anyway.)

If he is a US citizen, then he wasn't supposed to enter on VWP, but there isn't really any consequences for that. "Overstay" isn't an issue for a US citizen, but he won't be able to use VWP anymore (which he wasn't supposed to use anyway). Just make sure he uses a US passport to enter the US in the future.

If he isn't a US citizen, there isn't a way to extend his status. Extension of Status is not possible for people who entered on VWP. However, if you apply for Adjustment of Status for him to become a permanent resident, he can stay in the US for as long as it is pending.

Your son does not, as a matter of law, possess any nonimmigrant status (VWP or or otherwise), because, as others have explained, he is already a US citizen, and nonimmigrant status is only available to aliens.

He was admitted under the VWP as a matter of administrative convenience, because, I suppose, his US citizenship was not conclusively proven to the admitting immigration officer. But there is absolutely no way a US citizen can get a "black mark" for "overstaying" because it is not possible for a US citizen to overstay.

In support of this analysis, I offer 8 USC 1187, the section of the United States Code that controls the VWP. Note that the people concerned are denoted as "aliens" who meet certain criteria.

The applicable definition of "alien" may be found at 8 USC 1101(a)(3):

(3) The term “alien” means any person not a citizen or national of the United States.

1. He will never be entering the USA on the VWP program again, anyway, once he has his US Passport. Unless you are a multi-generation expat, your son is not "eligible" to be a US Citizen; he was a US citizen when he was born. (See, e.g., discussion of Ted Cruz's and George [not Mitt] Romney's citizenship for purposes of Presidential eligibility.)
2. Many types of status are extended if you have an application pending for change. I can't address whether that applies here since (a) I am not an immigration lawyer and don't know and (b) your children are citizens and the question is moot.