harry potter - How does the diary prove Voldemort is the Heir of Slytherin?

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

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.}