Dual nationality : ESTA and non ESTA citizen with one valid nonimmigrant USA visa

You can use either passport with no problem, however:

I recommend you apply for the ESTA with your French passport and transit the US with it.

The reason is this:

Visa Waiver Program travelers may use Automated Passport Control kiosks to clear immigration. This is usually much faster than waiting in the immigration queue with visa holders and first-time visitors. These kiosks are now available in most US airports' international arrival areas, and at all US preclearance airports. You simply swipe your and your daughter's passports in the machine, touch the screen when it asks you the customs declaration questions, take your receipt, have a much shorter immigration interview, and go directly to baggage claim.

Depending on which type of visa you have, (you didn't mention it) you might not be able to do this on the Honduras passport.

I have had a similar dilemma (minus a child and specific combination of countries). I simply did not want to travel with two passports, as the second is a pain to replace in case of any theft or emergencies (and that is where my visa was).

In the end, I hesitantly called ESTA customer service number (I thought I would have to be on hold forever).

I was told that admittance is based on the passport you choose to use for crossing the US border. Hence, if you choose to cross the border with a passport without a visa, but ESTA eligible, you are allowed (and of course required) to apply for ESTA. You have no obligation to travel with your "visa-containing" passport.

It depends in part on whether the nonimmigrant visa you hold actually authorizes you to transit the US. For example, if you hold a G visa, you probably cannot use that to transit the US on a personal trip unless you reside in the US.

Assuming you have a B1 and/or B2 visa, or an actual transit visa, or some other visa that is valid for your present purpose, then if I were you I'd travel on the Honduran passport. You already have the visa, so use it.

I would not worry about using a different country's passport than that your child is using. Bring your child's birth certificate to prove your relationship. Border guards see children traveling with their parents and using different passports all the time. I know this because pretty much every time I travel internationally, I see at least one family with passports from multiple countries (I live in the US, where there are many immigrant families). If I see such a sight at least once in the 10 to 30 minutes I spend in the security line and passport control line, how many times must the passport officers see it in each shift?

If the preceding paragraph did not convince you, you can also bring both your passports (which you should probably do anyway, as you'll want to use your French one to get back into France). If anyone should say, "how can you be this child's mother if you are not French?" you can respond by saying that you are French, and proving it with your other passport. But nobody is going to say that.

Here's a source that should reassure you that, according to the US, you can choose whether to travel on your visa as a Honduran or with ESTA as a Frenchwoman:

Do I need to apply for ESTA if I have dual citizenship from both a VWP country and a non-VWP country?

If you are flying under the terms of the VWP, with your VWP passport, you must apply for ESTA. If you are using your non-VWP country's passport, you will require a visa and therefore ESTA does not apply to you.

Source: https://help.cbp.gov/app/answers/detail/a_id/1094/~/do-i-need-to-apply-for-esta

Finally, you seem to be aware, but others who find this question might not be, that your decision about which passport to use is essentially independent from the fact that you're traveling with your French-citizen daughter, because she needs an independent ESTA even if you also have one:

Do I need to apply for ESTA if I am traveling with a minor (child under age 18) who is a citizen of a VWP country?

If the minor is traveling to the United States under the VWP as a tourist 90 days or less, he/she is required to obtain an independent ESTA approval.

Source: https://help.cbp.gov/app/answers/detail/a_id/1094/~/do-i-need-to-apply-for-esta

I would say that it's probably easier to get an ESTA and use your French passport. The reason is that then you will be travelling with the same nationality and circumstances as your daughter. Of course it's not illegal, or uncommon, for parents to travel with their children who have a different nationality. However, immigration agents usually inspect families more carefully if there is even the slightest hint that a child might not be accompanied by their real parents.

If you want to save the $14 ESTA fee, then you can choose to enter with the US using your Honduran passport if you like.