Could Neville have picked up the prophecy in the Ministry instead of Harry?

Question

Or to put it another way, why does the prophecy in the Ministry have a (?) sign in front of Harry's name?

This was the label on the prophecy

S.P.T to A.P.W.B.D
Dark Lord
and (?) Harry Potter  

Apparently Harry wasn't born when the prophecy was made, so if the prophecy was transcribed soon after when it was made (hence explaining the presence of the (?) sign), Harry's name couldn't have been there since the beginning. If it was transcribed after Voldemort disappeared, by then it would have been pretty clear that the prophecy meant Harry and there wouldn't have been the need for the (?) mark, but it's still there.

So, as far as the ball of the prophecy is concerned, could Neville have also picked it up safely?

Answer 1

Probably not

The prophecy originally did not mention Harry's name, or indeed Neville's. It merely mentioned a few characteristics of the individuals who could ensure Voldemort's downfall, such as their date of birth. The Keeper changed the labeling of the prophecy after Voldemort's attempt to kill Harry:

“But then . . . but then, why was it my name on the prophecy and not Neville’s?”

“The official record was relabeled after Voldemort’s attack on you as a child,” said Dumbledore. “It seemed plain to the keeper of the Hall of Prophecy that Voldemort could only have tried to kill you because he knew you to be the one to whom Sybill was referring.”

—Harry Potter and the Order of the Phoenix

We know that only the subjects of a prophecy can retrieve it:

“So he wanted me to come and get it, did he? Why?”

“Why?” Malfoy sounded incredulously delighted. “Because the only people who are permitted to retrieve a prophecy from the Department of Mysteries, Potter, are those about whom it was made, as the Dark Lord discovered when he attempted to use others to steal it for him.”

—Harry Potter and the Order of the Phoenix

Note the language here: are permitted. This seems to indicate that the protections are enchantments placed by the Ministry themselves, based on their understanding of whom the prophecies reference. As such, Neville could not have removed the prophecy unless he were included in the enchantments. It seems unlikely that he could have been, after the relabeling.

As to why there was a question mark before Harry's name, that was probably simple prudence. While it may have seemed clear to the Keeper that the prophecy referred to Harry, they probably would not want to make things seem certain when Harry was not mentioned in the prophecy itself.

Answer 2

If Lucius Malfoy is correct, and by the evidence of Broderick Bode it seems that he is,

[T]he only people who are permitted to retrieve a prophecy from the Department of Mysteries, Potter, are those about whom it was made, as the Dark Lord discovered when he attempted to use others to steal it for him.

Harry Potter and the Order of the Phoenix - p.693 - Bloomsbury - Chapter 35, Beyond the Veil

(See also Adamant's answer)

'Avery told me Bode would be able to remove it.'

'Bode could never have taken it, Master ... Bode would have known he could not ... undoubtedly, that is why he fought so hard against Malfoy's Imperius Curse ...'

Harry Potter and the Order of the Phoenix - p.516 - Bloomsbury - Chapter 26, Seen and Unforeseen

I think that there is no doubt that Neville could not have picked it up.

The crucial quote, for me, follows on from the quote Adamant provides:

'Then - it might not be me?' said Harry.

'I am afraid,' said Dumbledore slowly, looking as though every word cost him a great effort, 'that there is no doubt that it is you.'

'But you said - Neville was born at the end of July, too - and his mum and dad -'

'You are forgetting the next part of the prophecy, the final identifying feature of the boy who could vanquish Voldemort ... Voldemort himself would mark him as his equal. And so he did, Harry. He chose you, not Neville. He gave you the scar that has proved both blessing and curse.'

'But he might have chosen wrong!' said Harry. 'He might have marked the wrong person!'

'He chose the boy he thought most likely to be a danger to him.' said Dumbledore. 'And notice this, Harry: he chose, not the pure-blood (which, according to his creed, is the only kind of wizard worth being or knowing) but the half-blood, like himself. He saw himself in you before he had ever seen you, and in marking you with that scar, he did not kill you, as he intended, but gave you powers, and a future, which have fitted you to escape him not once, but four times so far - something that neither your parents, nor Neville's parents, ever achieved.'

Harry Potter and the Order of the Phoenix - p.742 - Bloomsbury - Chapter 37, The Lost Prophecy

The Prophecy is quite explicit.¹

the Dark Lord will mark him as his equal,

This was Harry, not Neville.

but he will have power the Dark Lord knows not

Well Neville appears to have had that power too, the power of love, to a great extent. But it was certainly Harry who walked into that Hall of Prophecy with his mother's charm protecting him. And Harry was certainly shown to be somewhat unique in the depth of his love, and in its endurance:

'You are protetcted [from the desire to become one of Voldemort's followers], in short, by your ability to love!' said Dumbledore loudly. 'The only protection that can work against the lure of power like Voldemort's! In spite of the temptation you have endured, all the suffering, you remain pure of heart, just as pure as you were at the age of eleven, when you stared into a mirror that reflected your heart's desire, and it showed you only the way to thwart Lord Voldemort, and not immortality or riches. Harry, have you any idea how few wizards could have seen what you saw in that mirror?

Harry Potter and the Half-Blood Prince - p.478 - Bloomsbury - Chapter 23, Horcruxes

Would Neville have shown such love? Would Neville's mother have died for him and given him an unbreakable protection? Possibly. But it doesn't matter. Because she didn't, did she? It didn't happen to Neville, did it? The Prophecy tells of someone who will have a power that the Dark Lord knows not, and a part of that seems to be his mother's charm, and Neville doesn't have that, whether he could have had, or would have had is immaterial, he does not have.

And in any case, the Dark Lord will mark him as his equal. The Prophecy foretells Voldemort picking a boy, and that boy will be given the power to defeat him. Voldemort picks a boy and gives that boy the power to defeat him. That boy is not Neville.

¹Harry Potter and the Order of the Phoenix - p.741 - Bloomsbury - Chapter 37, The Lost Prophecy

Answer 3

Although not 100% reliable the HP Wiki tells us that the Unspeakables made the change to the tag after Voldemort's attack.

Ron, however, found the Prophecy with Harry's name on it on row 97. It read:

S.P.T to A.P.W.B.D

Dark Lord

and (?) Harry Potter

S.P.T are the initials of Sybill Patricia Trelawney, and A.P.W.B.D are the initials of Albus Percival Wulfric Brian Dumbledore. A question mark (?) was placed on the prophecy because at that time, the identity of the boy who would defy Lord Voldemort was still not entirely certain. When, however, Voldemort's downfall was brought upon after his attack on Harry Potter, the record of the prophecy was changed by the employees of the Department of Mysteries (known as Unspeakables) so that Harry's name was shown.

So what I am gathering is the prophecy only had the question mark on it until Voldemort made his attack. The Ministry added "Harry Potter" at a later time. So, I do not think Neville would have been able to pick it up because the confusion had gone away. The prophecy became to be about Harry by Voldemort's own doing.