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Voldemort was said to be very interested in the four founders of Hogwarts. He created a Horcrux from the Ravenclaw diadem, the Hufflepuff cup, and the Slytherin locket. Did he try to find a Gryffindor relic to complete the set, or did he not bother?

I can't see him giving up on such a task, but maybe there was insufficient time.

Gryffindor's sword was hiding 'inside' the Sorting Hat, but a famous family usually has some other heirlooms too.

If there are no canon answers I'd be interested in your own speculation!

From Dumbledore himself in HBP, while discussing Voldemort's horcruxes with Harry:

"I am confident, however, that the only known relic of Gryffindor remains safe."

_{^{HBP, Chapter 23, p505, Scholastic Edition. Emphasis added}}

So, I believe we can take this as confirmation that no other family has another heirloom. Else, Dumbledore and other magical historians (like Bagshot) would've known about it.

And, as we know, only Gryffindors can take the sword from the Sorting Hat. Voldemort, as a Slytherin, wouldn't be able to even if he wanted; he's really better off giving up since he is unable to acquire it by himself. Though he may be able to trick a Gryffindor into getting it from the Hat for him, remember that Voldemort likes to operate alone, especially when it comes to his horcruxes. So I don't think that plan would appeal to him.

(It's worth noting too that the Sorting Hat is a relic of Gryffindor's. I don't know if it can be turned to a Horcrux tho, given its current enchantment.)

I think the answer is simply that it would be impossible to protect, seeing as ANY Griffindor could pull the sword out of the hat. No matter what protections he put on it, any one of his enemies who was in Griffindor would have easy access. It would never be safe.

Even though he probably didn't realize it, he did have a Horcrux in each house.

• The Locket/Slytherin.


• The Cup/Hufflepuff.


• The Diadem/Ravenclaw.


• 
  Harry Potter/Gryffindor.

Voldemort would have most likely treasured this as a horcrux due to the fact that it is indestructible hence it being most likely if a Gryffindor did happen to receive it would not be able to destroy it therefore having the possibility of making himself truly immortal even though it made contact with the basilisk venom since it makes the blade more powerful it also says to DESTROY the horcrux if you aren't able to destroy the sword of Gryffindor then there would be the possibility of not destroying the horcrux

Gryffindor is his enemy, so it's probably too risky to put a part of his soul in his enemy's hands.

If he obtained it and hid it, the Gryffindors will go searching for it.

That's my speculation.

I think the answer here is Dumbledore himself. While there are a variety of reasons that Voldemort shouldn't/couldn't have had access to the sword, Rowling went to blatantly obvious lengths to make it known that he was the only wizard that Voldemort feared, despite Dumbledore's mediocre performance in the Ministry of Magic.

It was too much of a risk for Voldemort to make a horcrux out of one of the items most closely guarded by Dumbledore. Dumbledore and Voldemort were both brilliant wizards and Voldemort had to know that even if Dumbledore didn't flat out destroy the sword, it would be an instant "give" to the Horcrux situation.

This is all, of course, without the knowledge that Dumbledore would fall under the spell of the Slitherin ring. Even if Voldemort had known that the ring would be a temptation, that would have been further reason not to put another horcrux that closely in his presence.

To put it simply, Voldemort was smart to stay out of Dumbledore's way. It ultimately didn't matter, but if it did, the HP series would have had a much different outcome, wouldn't it?