This question is more complex than it may appear. There seems to be consensus that a singular verb should be used in formal writing whenever the subject of a sentence is more than one [noun], or at least that this is (much) better than ?there are more than one. I subscribe to this.
It does not matter how many things the writer might expect there to be in reality: it is always if there is more than one species.
Nor does it matter what noun comes after one. It is always is; the word one forces a singular verb without apparent exception.
But why does this at all surprise us?
We are puzzled by this construction because the subject does not agree with the verb—at least not if analyzed according to conventional grammar. Consider the following sentence:
There are more men in the room.
Is would be impossible. The sentence is easy enough to analyse:
- more men = subject
- are = finite verb
No problem there: subject and verb agree. More men is the subject, or at least the head of the subject.
There are more men than just John in the room.
The core of the syntax remains the same; the core of the subject is still more men. The addition than x is either an elliptical clause or a prepositional phrase that is part of the subject, depending on your model; in any case, than x is not what determines whether it is are or is. More men are is the core of the sentence.
There is more than one man in the room.
Suddenly the verb changes. Has the core syntax of the sentence changed? No: for the sake of consistency, we must say more is the subject and is the finite verb. The phrase than x is still not the head of the subject, no more than in the first sentence (there are more men than just John). If more is the head, then it must be elliptical, since it is only an adjective: more of what? If we hypothetically supply the omitted noun, we get:
*There is more [men] than one man in the room.
There is no other word that we could fill in, though of course this is wrong: *there is more men is both unidiomatic in this register and in violation of the rule that subject and verb must agree.
Then what causes this singular is in there is more than one man? The phrase than x should not determine the number of the verb: and yet it does. That is why this construction is idiomatic, as opposed to regular: it violates the rule that subject and verb must agree. But it is by all means "correct". That is what idiom is: a widely accepted phrase that violates the regularity of our language. However some of us might like it to be, language is just never regular in all respects; this bit of idiom happens to have triumphed over regularity and is now the norm. Idiom must be judged case by case and often varies across registers and dialects.
But could this disagreement of subject and verb be explained away by other factors? Let's see what I can come up with.
It could be that the somewhat fixed phrase there is is what does it. But that phrase could not explain singular is in this sentence:
More than one man is still in the house.
Could this is be explained by the immediate precedence of one man? It is conceivable that the singular number of one man leads us to an anacoluthon in the next word is: we see a singular number and noun, and we cannot resist the pressure of proceeding with a singular verb. But then this phenomenon should not occur if the verb came before the subject:
Not only has more than one man been seen near the power plant, but...
*Not only have more than one man been seen near the power plant, but
It seems clear that have would be wrong, even more so than in the previous sentence patterns. So whether the verb comes before or after doesn't matter.
How can this oddity of disagreement be explained? If we look at it reductionistically, in terms of association and pattern recognition as they occur in the brain, I suspect that the word one exerts such an enormous influence on our perception of a sentence that it overrules more, despite the ordinarily forcing rule of agreement; it does so even despite the sense of multitude inherent in the phrase more than one man as a whole, which must always refer to multiple objects in reality. When we write one man, we have the image of one man at an irresistibly prominent place in our working memory. (Other, somewhat similar idioms exist, so I don't believe this to be a unique situation.)
We don’t have ‘make enemies with’, but we do have ‘make enemies of’, just as we have ‘make friends of’, so in ‘friends with’, ‘friends’ would appear to be a noun. (The OED has ‘friends with’ under its definitions of ‘friend’ as a noun.) ‘Friends’, because friendship requires two participants as a minumum. It’s not just friends, of course. We can be ‘pals with’, ‘buddies with’, ‘mates with’, ‘chums with’, ‘partners with’ . . .
Best Answer
Actually, this construction seems to be be attested in some documents indexed by Google Books:
(The federal reporter - Volume 219 - Page 170)
(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.
"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