"なる" means "become" and takes "に". "What did you want to become when you were a kid?" I like to think of it as "turn (in)to" to remember the "ni" part of it. "子供は大人になった" "The child became (turned into) an adult." "何が" would do weird "What wanted to become..." things to your sentence, which I think are ungrammatical.

as you know that.. in Japanese there is a conflict and lots of misunderstanding for using particles like " が" and " は" .Basically " が" or "は" is a subject particles. But sometime "が" is also use to emphasis your dialogue. like " すみませんが"。 Here in your sentence "子供のとき、何になりたかったですか。" 何 is actually a character and " に" particle stands for the existence of that character. in other words " in your childhood you want to be like someone or you may say that you want the existence of someone in you. As we know that the "に" is a existence particle like," 机の上に" here "に" indicates that something is existing on the table. So i thing that why in your composition " 何に"is preferable against " 何が。