So, I reckon Shirley Temple is the most famous child actor, and my daughter is nominating Macaulay Culkin for the spot. There is serious cred hanging here…
Start by insisting on a definition of how to measure “famous”. That should reveal the fatuity of the argument.
You also need to clarify your question by asking “most famous when?”. Most famous at the peak of their fame or most famous now.
You’re both wrong, it’s Danny Cooksey.
Actually, in all seriousness, I think Miley Cyrus might be in the running too.
This is always going to be generational. I mentioned Shirley Temple in the office the other day and one of my colleagues (aged 25) had never heard of her.
How about Judy Garland? Daniel Radcliffe?
Shirley Temple was the first name to pop into my mind, but if we’re looking at more modern eras, I’d say the Olsen Twins, Daniel Radcliffe, Emma Watson, or Finn Wolfhard ought to be in the running.
Mickey Rooney?
Ron Howard?
I agree that it’s a generational thing. My first thought was Jay North followed by Ron Howard (1960s). But Shirley Temple (1930s) has them beat.
Really? 
I think of him as a fat old man.
Rooney first performed in vaudeville as a child actor, and made his film debut at the age of 6 years. He played the title character in the popular “Mickey McGuire” series of 78 short films, from age 7 to 13.
I thought of Shirley Temple. I’m 47, far too young to have actually watched any of Shirley Temple. I wouldn’t have thought of Macaulay Culkin, to me he just seems like a kid who was in a movie or two.
I’m in my mid-thirties. If I had to mention Daniel Radcliffe, Emma Watson, or Ron Howard in conversation I might feel I had to give a reference for who they are. Everybody knows who Shirley Temple was, even people who never saw one of her films.
Your 25 year old colleague was just having a bad day.
Most famous child star? Shirley Temple, no doubt.
Most famous child actor? Not as clear cut.
We also need to define “child actor”. Ron Howard or Emma Watson, for instance, both got their starts as children, but continued to have quite successful careers as adults. Do they count as “child” actors? Or Miley Cyrus was certainly a successful child celebrity, and she did do some acting, but she was much more known for her singing (even her acting was mostly playing a singer who was basically just herself, for the sake of showing off her singing). Does she count as a child “actor”?
Alfalfa, Spanky, Buckwheat, and Stymie were the first ones who came to my mind
Those are child roles, not child actors. If you can’t even remember the actors’ names, then you can scarcely say that they’re famous.
I need to know their names?!? 
I’d recognize them anywhere!
Jackie Coogan, famous for his child role in the Charlie Chaplin film “The Kid”.
He turned into an…interesting-looking adult.
Definitely Shirley Temple based on the number of years, the percent of the population that recognized her and the number of products and styles that she inspired. She was an American institution even as an adult. A close second is the Dione Quintuplets.
Deanna Durbin was far more famous than Miley Cyrus.