How does the diary prove Voldemort is the Heir of Slytherin?

Question

Harry: "The diary wasn't that special."
Dumbledore: "The diary, as you have said yourself, was proof that he was the Heir of Slytherin; I am sure that Voldemort considered it of stupendous importance."

^{— Harry Potter and the Half-Blood Prince, Chapter 23: Horcruxes (slightly more than halfway through)}

How does the diary prove that? Sure, it contains a fragment of his soul which can come out and open the chamber, but wouldn't the same be true of any other Horcrux, as well as by Voldemort being able to do it himself? How does the diary lend any additional credence to the claim?

Every other Horcrux was a rare magical item with a long history, but the diary was . . . just a diary. As far as I know the books don't even make it clear where it came from or how Voldemort got it. It was an ordinary notebook. So Dumbledore's argument for why it was chosen, instead of something more special, makes little sense. How is it different from any other object he might have chosen, like a pencil sharpener or a can of WD-40?

Answer 1

Riddle probably wrote in it

The original quote from Chamber of Secrets might seem to imply that Voldemort made the diary into a Horcrux for the express purpose of opening the Chamber.

"Well, he certainly kept an annoyingly close watch on me after Hagrid was expelled,” said Riddle carelessly. “I knew it wouldn’t be safe to open the Chamber again while I was still at school. But I wasn’t going to waste those long years I’d spent searching for it. I decided to leave behind a diary, preserving my sixteen-year-old self in its pages, so that one day, with luck, I would be able to lead another in my footsteps, and finish Salazar Slytherin’s noble work.”

—Harry Potter and the Chamber of Secrets

This may be. However, Voldemort may have purchased the diary before opening the Chamber in his fifth year:

Harry saw at once that it was a diary, and the faded year on the cover told him it was fifty years old. He opened it eagerly.

—Harry Potter and the Chamber of Secrets

This does not mean that it was literally fifty years old, merely approximately so. In any case, even if he purchased it the year he opened the Chamber, he may still have written in it.

Indeed, Harry explicitly says that Riddle wrote the diary:

“It was this diary,” said Harry quickly, picking it up and showing it to Dumbledore. “Riddle wrote it when he was sixteen. . . .

—Harry Potter and the Chamber of Secrets

The purpose of the diary was ultimately to open the Chamber once more:

"Well, he didn't want his hard work to be wasted," said Harry. "He wanted people to know he was Slytherin's heir, because he couldn't take credit at the time."

—Harry Potter and the Half-Blood Prince

It is this statement that Dumbledore references when he says that the diary proved that Riddle was Slytherin's heir. Nonetheless, I suspect Riddle originally wrote in the diary about his opening the Chamber of Secrets: it is the only interpretation that makes sense.

It seems most likely that Riddle wrote in his diary the details of how he had opened the Chamber. Thus, the diary proved that Riddle had opened the Chamber before he made it into a Horcrux. After he made it into a Horcrux, the piece of soul inside could control the ink in its pages (and thus presumably kept the account hidden).

_{Alternately, perhaps Riddle made the diary precisely because he needed a Horcrux that could communicate with people directly, which the diary could do by means of the ink spilled on its pages (whereas presumably the diadem could not). As such, proving that he was the Heir of Slytherin and making the diary in particular into a Horcrux may have been essentially the same goal. The locket did communicate, after a fashion, but only with Ron Weasley, who had perhaps been a bit too emotionally influenced by it.}

Answer 2

As explained by Binns,

“Sir — what exactly do you mean by the ‘horror within’ the Chamber?” “That is believed to be some sort of monster, which the Heir of Slytherin alone can control,” said Professor Binns in his dry, reedy voice.
— _{^{Harry Potter and the Chamber of Secrets}}

The diary was able to make Ginny open the Chamber and call the basilisk

Riddle opened his mouth wide and hissed — but Harry understood what he was saying… “Speak to me, Slytherin, greatest of the Hogwarts Four.”
— _{^{Harry Potter and the Chamber of Secrets}}

And it confessed being Lord Voldemort:

“Voldemort,” said Riddle softly, “is my past, present, and future, Harry Potter…” He pulled Harry’s wand from his pocket and began to trace it through the air, writing three shimmering words: TOM MARVOLO RIDDLE Then he waved the wand once, and the letters of his name rearranged themselves: I AM LORD VOLDEMORT
— _{^{Harry Potter and the Chamber of Secrets}}

So yes, the diary was a complete proof that Voldemort was a Heir of Slytherin.

Voldemort himself would have been able to prove it too by opening the Chamber himself, but at that moment he was unable to do so (he was a shadow in Albania).

So the diary was just a backup he had in case he was unable to do so, just exactly it happened. That's why he gave it to Lucius:

Lucius did not know what the diary really was. I understand that Voldemort had told him the diary would cause the Chamber of Secrets to reopen because it was cleverly enchanted.
— _{^{Harry Potter and the Half-Blood Prince}}

Therefore, the diary was some kind of exception: it wasn't important because of what it had been, it became important because of what it become once being a Horcrux. If Voldemort had chosen another object, then the diary wouldn't had became important.

But Voldemort chose the diary as the future Horcrux capable of opening the Chamber of Secrets and proving that Voldemort was the Heir of Slytherin. That's the same as what happened with Harry Potter, who became the protagonist because Voldemort chose him:

“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.
— _{^{Harry Potter and the Order of the Phoenix}}

Answer 3

Short answer:
The memory Riddle, enclosed in the diary, says so:

"Now, Harry, I'm going to teach you a little lesson. Let's match the powers of Lord Voldemort, Heir of Salazar Slytherin, against famous Harry Potter, and the best weapons Dumbledore can give him..."
(Chamber of Secrets, Chapter 17, The Heir of Slytherin)

Longer answer:

1. Only the heir of Slytherin can open the Chamber of secrets

The story goes that Slytherin had built a hidden chamber in the castle, of which the other founders knew nothing. Slytherin, according to the legend, sealed the Chamber of Secrets so that none would be able to open it until his own true heir arrived at the school. The heir alone would be able to unseal the Chamber of Secrets, unleash the horror within, and use it to purge the school of all who were unworthy to study magic."
(Chamber of Secrets, Chapter 9, The writing on the wall)

2. The diary holds Riddle's memories of his 5th year of school, when he opened the Chamber of Secrets

Harry saw at once that it was a diary, and the faded year on the cover told him it was fifty years old. He opened it eagerly. On the first page he could just make out the name "T M. Riddle" in smudged ink.

(Chamber of Secrets, Chapter 13, The very secret diary)

[...]

He waited eagerly for Riddle's reply.

"Lucky that I recorded my memories in some more lasting way than ink."

[...]

"I mean that this diary holds memories of terrible things."

[...]

"Are you a ghost?" Harry said uncertainly.

"A memory," said Riddle quietly. "Preserved in a diary for fifty years.
He pointed toward the floor near the statue's giant toes. Lying open there was the little black diary Harry had found in Moaning Myrtle's bathroom.

[...]

"I knew it wouldn't be safe to open the Chamber again while I was still at school. But I wasn't going to waste those long years Id spent searching for it. I decided to leave behind a diary, preserving my sixteen-year-old self in its pages, so that one day, with luck, I would be able to lead another in my footsteps, and finish Salazar Slytherin's noble work."

(Chamber of Secrets, Chapter 17, The Heir of Slytherin)

=> Hence, the memories in the diary show that Riddle is the Heir of Slyterin. And since Tom Marvolo Riddle is Lord Voldemort, the diary proves that Voldemort is the heir of Slytherin.