I have a one-year old son. He was born in Russia, but has already an American passport. He also currently lives in Russia. Does I need to fill out an electronic authorization form for him to travel to Mexico. Thank you for your assistance.
No. As a US citizen, your son has visa-free access to Mexico for up to 180 days. See Visa policy of Mexico for more information.
If your son has any other citizenships (like Russian for example, but you didn't say either way), that doesn't matter if he enters Mexico using a US passport.