Any answer to this would be speculation, since we're unlikely to have a definite reason for such a minor detail.
There are a few possible explanations:
1. It was the axe used on <PERSON>. Perhaps he kept it as a souvenier (he was known to celebrate his death, so it's not impossible).
"Ah," <PERSON> waved an elegant hand, "a matter of no importance... It's not as though I really wanted to join.. Thought I'd apply, but apparently I 'don't fulfill requirements' –"
In spite of his airy tone, there was a look of great bitterness on his face.
"But you would think, wouldn't you," he erupted suddenly, pulling the letter back out of his pocket, "that getting hit forty-five times in the neck with a blunt axe would qualify you to join the Headless Hunt?"
(The Deathday Party, The Chamber of Secrets)
1. It was used for an execution. There is precedent for this in the books:
Before <PERSON> could answer, two wizards came through the castle doors behind him. One was so ancient he appeared to be withering before their very eyes; the other was tall and strapping, with a thin back mustache. <PERSON> gathered that they were representatives of the Committee for the Disposal of Dangerous Creatures, because tie very old wizard squinted toward <PERSON>'s cabin and said in a feeble voice, "Dear, dear, I'm getting too old for this... Two o'clock, isn't it, <PERSON>?"
The black-mustached man was fingering something in his belt; <PERSON> looked and saw that he was running one broad thumb along the blade of a shining axe. <PERSON> opened his mouth to say something, but <PERSON> nudged him hard in the ribs and jerked her head toward the entrance hall.
(Professor Trelawney's Prediction, The Prisoner of Azkaban)
1. 773
It was part of one of the nasty incidents caused by the Death Eaters while they were at Hogwarts.
"Let us say that I did not take it for granted that he was trustworthy," said <PERSON>. "I had, as I have already indicated, resolved to keep a close eye upon him, and so I did. I cannot pretend that I gleaned a great deal from my observations at first. He was very guarded with me; he felt, I am sure, that in the thrill of discovering his true identity he had told me a little too much. He was careful never to reveal as much again, but he could not take back what he had let slip in his excitement, nor what Mrs. <PERSON> had confided in me. However, he had the sense never to try and charm me as he charmed so many of my colleagues.
"As he moved up the school, he gathered about him a group of dedicated friends; I call them that, for want of a better term, although as I have already indicated, <PERSON> undoubtedly felt no affection for any of them. This group had a kind of dark glamour within the castle. They were a motley collection; a mixture of the weak seeking protection, the ambitious seeking some shared glory, and the thuggish gravitating toward a leader who could show them more refined forms of cruelty. In other words, they were the forerunners of the Death Eaters, and indeed some of them became the first Death Eaters after leaving Hogwarts.
"Rigidly controlled by <PERSON>, they were never detected in open wrongdoing, although their seven years at Hogwarts were marked by a number of nasty incidents to which they were never satisfactorily linked, the most serious of which was, of course, the opening of the Chamber of Secrets, which resulted in the death of a girl. As you know, <PERSON> was wrongly accused of that crime.
(A Sluggish Memory, The Half-Blood Prince) 573