While rereading the entire Harry Potter series I noticed that the number 7 occurs quite often, for instance:
Did J.K. Rowling ever mention why 7 was so important in Harry Potter?
“Yes, sir,” said Riddle. “What I don’t understand, though — just out of curiosity — I mean, would one Horcrux be much use? Can you only split your soul once? Wouldn’t it be better, make you stronger, to have your soul in more pieces, I mean, for instance, isn’t seven the most powerfully magical number, wouldn’t seven — ?”
“Merlin’s beard, Tom!” yelped Slughorn. “Seven! Isn’t it bad enough to think of killing one person?"
-- HP and the Half-Blood Prince
As this answer says, seven is considered a magical number in real-life legends and folklore, but here is confirmation that seven has magical significance in-universe.
The magical properties of the number seven were first discovered by the 13th-century Arithmancer Bridget Wenlock:
Wenlock first scribbled down her ground-breaking theorem on the magical properties of the number seven at breakfast one day apparently on the back of an envelope, using her usual invisible ink. She then proceeded to send her cousin a letter, using what she later believed to be the very envelope concealing the theorem.
-- Miranda Goshawk's Book of Spells
This is also confirmed in the Hufflepuff welcome letter at Pottermore:
Bridget Wenlock, the famous thirteenth-century Arithmancer who first discovered the magical properties of the number seven
7 is a magic or prominent number in folklore, and is also important in many religions. The 7th son of a 7th son is said to have magical qualities. This is not a concept J.K. Rowling came up with nor is it exclusive to the Harry Potter books.
The number 7 is practically everywhere in mainstream culture. It is referenced as a lucky number, we play '7 boom', we are most likely to choose it as a number between 1 and 10, etc.
If you'd like to read more about 7 in Harry Potter, here's a link to the Harry Potter wiki page on the topic.
Again, 7 is considered a magical number in fairy tales and folklore, not just Harry Potter. Even Tom Riddle's statement:
I mean, for instance, isn’t seven the most powerfully magical number, wouldn’t seven — ?
Would have been based on real-world beliefs regarding powerful numbers.
As per Wizard of the Month articles,
The Arithmancer Bridget Wenlock (1202–1285) was the first witch to establish the magical properties of the number seven.
If you read the book carefully, you will realise Dumbledore and Mrs Weasley telling Harry that 7 was a powerful magic number in the magical society. Riddle (Voldemort) split his soul into 7 because it was a lucky and magical number. If you encounter the conversation between the old potions master and Riddle in the memory of the old potions master, you would read that Riddle asked what would happen if he split his soul into 7.
I agree with DVK that 7 is the most magical number in Potterverse because canon (i.e. Tom Riddle) says so.
According to Miller's Law - The Magical Number Seven, Plus or Minus Two, seven is the approximate amount of single items a person is able to retain in their working memory at any one time. It could be that seven was a number of hiding locations that Voldemort could most easily remember, thus seven Horcruxes.
I ran ‘Tom Marvolo Riddle' through The Sorcerer's Companion Arithmancy Calculator and got these numbers:
Character Number: 7
Heart Number : 6¹
Social Number: 1²
Perhaps Tom Riddle found value in his Arithmancy chart. Remember he immediately believed Dumbledore when he was eleven and Dumbledore told him he was a wizard; he outright stated that he had known he was special all along. If seven - the most magical number - came up on his Arithmancy chart, he may have been eager to see this as a reinforcement of his belief that he was the most magical being in the world, and subsequently he would protect his life with seven Horcruxes.
A 7 means: Perceptive, understanding, and bright, sevens enjoy hard work and challenges. They are often serious, scholarly, and interested in all things mysterious. Originality and imagination are more important than money and material possessions. Sevens can also be pessimistic, sarcastic, and insecure.
There are seven years of magical schooling through Hogwarts, when witches and wizards really come into their own, increase their powers, learn to control magic, and prepare to go out into the world and do great things ("Terrible! But great. . .")
¹ I'm not even going to pretend that Voldemort has a heart.
² This fits Voldemort as well. A 1 means: This is the number of the individual, the solitary unit. Ones are independent, focused, and determined. They set a goal and stick to it. They are leaders and inventors. Ones find it difficult to work with others and don't like to take orders. They can be self-centered, egotistical, and domineering. They are often loners.
In addition to what others have pointed out, I'd like to point out that 7 appears all over in religion and popular culture.
One really obvious example is the fact that there are 7 days in a week (which actually comes from the days of creation in the book of Genesis as well as the Sabbath in the 10 Commandments). This probably has a lot to do with why it's considered the "number of completion."
A few other notable examples are the "Seven Deadly Sins" and "The Seven Habits of Highly Effective People."
This is also the same number as the number of books in C. S. Lewis's Chronicles of Narnia, which itself appears to correspond to the medieval viewed that there were 7 planets: to the Jupiter, Mars, the Sun, the Moon, Mercury, Venus, and Saturn.
This has made its way into popular culture. For example, consider this verse from a Muddy Waters song (I'm Your Hoochie Coochie Man):
On the seventh hour
On the seventh day
On the seventh month
The seventh doctor say
"He was born for good luck
And that you see
I got seven hundred dollars
And don't you mess with me
This is hardly the only example of this in popular music. As another example, from Stevie Wonder's Superstition:
Very superstitious, writings on the wall
Very superstitious, ladders bout' to fall
Thirteen month old baby, broke the lookin' glass
Seven years of bad luck, the good things in your past
This is, of course, referring to popular superstitions.
With that said: the H.P. books explicitly state that Voldemort believed that 7 was a lucky number, but this is merely reflective of a much wider cultural set of beliefs.
