What's wrong with "One of my children's name IS John"?

Actually, this construction seems to be be attested in some documents indexed by Google Books:

The Witness: One of my children's name is Richard White.

(The federal reporter - Volume 219 - Page 170)

Yes, but prior to that time it was in one of my children's name, prior to that time, I think [...]

(Records and Briefs in Cases Decided by the Supreme Court of Minnesota, 1907)

Of course, this is just an interesting fact; by itself, it doesn't answer your question. These could be simple production errors (which are common in speech), or it could be that the speakers quoted here had internalized different grammatical rules than you did. (Or these could even be transcription errors!) We still have to explain why you find this sort of sentence unacceptable.

Unfortunately, I don't know the relevant rule making it ungrammatical. But I do think I can explain why you judge sentence 2 to be grammatical.

I think in your sentence 2, the word "one" is, or at least seems like it can be, interpreted to refer to a bed rather than a child.

2: It's actually one of my children's, but you can sleep in this bed.

"This bed is actually one of [my children's beds], but you can sleep in it"

This means much the same thing as a hypothetical "*This bed is [one of my children's] bed" would, so it's hard to notice the difference in implied structure when only the elided form is used. However, I think the second structure would actually be ungrammatical, just as you say "*One of my children's name is John" is ungrammatical.

At first, I thought it might have something to do with the indefiniteness, but "a child's name" is acceptable, and "the youngest of my children's name" doesn't seem fully acceptable (although I might be imagining that it sounds slightly better than "one of my children's name").

Personally, I feel a bit uneasy with using 's-genitives after phrases that end in plural nouns no matter what the internal structure of the noun phrase is. "The queen of England's crown" sounds OK to me, but "The queen of the mice's crown" less so. I think a relevant point is whether "The father of my children's name" sounds any more or less acceptable to you.

I think the "clitic" nature of the English possessive -'(s) construction is somewhat exaggerated in the kind of short explanations that we provide on this site: there are definite complications and restrictions concerning its use (which I know have been analyzed in linguistic literature, but I am not familar enough with it to say more than that). Wikipedia has a short overview of some relevant analyses: Status of the possessive as a grammatical case.

"Also, here is an interesting paper I found: The English “Group Genitive” is a Special Clitic", Stephen R. Anderson

My musing: We have inventory of possible constructs we expect to hear after a sentence-initial "One of my brothers'" (or, indeed, "One of my children's"). But the usage you're asking about isn't common, so it isn't part of that inventory, or else it's in the inventory but assigned a low priority. After hearing or reading "One of my children's", we have already mapped out in our heads a likely structure for the sentence, a "garden path" situation. When the next word doesn't fit that structure, it leads to confusion and dissonance.

Therefore, it's better to stick with "The name of one of my children is John" or "One of my children is named John".