What is the best and cheapest nearby country to visit from Koh Samui for a 30-day visa extension with a baby?

They are staying at Koh Samui.

Gemma, what is your present location ?

Go to Siem Reap. Easy, short, direct flight. Get to see Angkor Wat while you are there. Small easy airport to navigate with a baby. Short 10-15 min drive airport to hotel with no traffic. Not expensive. Win win win win win.

Cambodia and Vietnam are also full page visas, and you have to use a service to get an invitation letter for Vietnam (cheap if you do it in advance). Phenom Phen is one of the sketchiest capital cities in SE Asia. If you will just fly out and back in for a 30 day visa waiver, you could literally take the cheapest round trip fare...

Any neighboring country will work and probably be comparably cheap. Vientiane Laos is generally considered an easy embassy to deal with, but is often crowded and Laos has a full page visa (takes up precious space in a crowded passport). Cambodia, Malaysia, and Vietnam all have their charms, so it is kind of up to what you want to do for 3 or so days while you go through the process. Bali is absolutely the most low traffic Embassy, and you would likely not spend more than 5 minutes dropping off and picking up, but Bali is more expensive for the flights... Worth it in my opinion for a 3 day place to chill, but it is substantially more expensive than any of the other spots above.

You can only get an Extension of Stay at your local Immigration Office. I guess you mean you like to fly to neighbor country and come back using the 30 day Visa Exempt Entry. Nearest country is depending on your location.