Harry has the Invisibility Cloak, one of the three most powerful objects in the wizarding world.
With time, more and more people learn the fact that Harry has it, however, nobody really cares. Shouldn't that be a big deal?
The Death Eaters went so far for the Elder Wand, did none of them know that Harry also possessed a Deathly Hallow?
In Chapter Twenty-Two of Deathly Hallows we see Harry thinking about Voldemort going after the Hallows:
Voldemort had been raised in a Muggle orphanage. Nobody could have told him The Tales of Beedle the Bard when he was a child, any more than Harry had heard them. Hardly any wizards believed in the Deathly Hallows. Was it likely that Voldemort knew about them?
Harry gazed into the darkness.... If Voldemort had known about the Deathly Hallows, surely he would have sought them, done anything to possess them: three objects that made the possessor master of Death? If he had known about the Deathly Hallows, he might not have needed Horcruxes in the first place. Didn’t the simple fact that he had taken a Hallow, and turned it into a Horcrux, demonstrate that he did not know this last great Wizarding secret?
Which meant that Voldemort sought the Elder Wand without realizing its full power, without understanding that it was one of three... for the wand was the Hallow that could not be hidden, whose existence was best known... The bloody trail of the Elder Wand is splattered across the pages of Wizarding history...
So it seems that Voldemort was unaware of the Hallows. This is confirmed by Dumbledore in Chapter Thirty-Five of Deathly Hallows:
“And Voldemort never knew about the Hallows?”
“I do not think so, because he did not recognize the Resurrection Stone he turned into a Horcrux. But even if he had known about them, Harry. I doubt that he would have been interested in any except the first. He would not think that he needed the Cloak, and as for the stone, whom would he want to bring back from the dead? He fears the dead. He does not love.”
This latter quote also tells us that even knowing about the Hallows wouldn't necessarily make someone interested in them, since aside from the Elder Wand they aren't necessarily so useful.
While the above quotes speak about Voldemort, we have no reason to assume that the Death Eaters were more aware of the Hallows than Voldemort. In fact, contrary to the assertion in the question here, it was Voldemort and not the Death Eaters who went after the Elder Wand.
Additionally, Harry's possession of the Invisibility Cloak was not such public knowledge. In Deathly Hallows the Death Eaters knew that he had the Cloak:
"Accio Cloak!" roared one of the Death Eaters.
Harry seized his folds, but it made no attempt to escape. The Summoning Charm had not worked on it. "Not under your wrapper, then, Potter?" yelled the Death Eater who had tried the charm and then to his fellows.
"Someone's there," came a rough whisper close at hand. "He's got an Invisibility Cloak. Could it be –?"
Yet as late as Chapter Six of Half-Blood Prince Harry was not sure if the Ministry knew that he had a Cloak:
He had stowed his Invisibility Cloak in his backpack and felt that, if that was good enough for Dumbledore, it ought to be good enough for the Ministry, though now he came to think of it, he was not sure the Ministry knew about his cloak.
Additionally, even if the Death Eaters knew about the Hallows, and knew that Harry had the Invisibility Cloak, they still wouldn't necessarily know that it was the Invisibility Cloak. After all, as explained by Xenophilius Lovegood in Chapter Twenty-One of Deathly Hallows, there are other Invisibility Cloaks besides the one that it is a Hallow:
"Mr. Lovegood," Hermione began again, "We all know that there are such things as Invisibility Cloaks. They are rare, but they exist. But –"
"Ah, but the Third Hallow is a true Cloak of Invisibility, Miss Granger! I mean to say, it is not a traveling cloak imbued with a Disillusionment Charm, or carrying a Bedazzling Hex, or else woven from Demiguise hair, which will hide one initially but fade with the years until it turns opaque. We are talking about a cloak that really and truly renders the wearer completely invisible, and endures eternally, giving constant and impenetrable concealment, no matter what spells are cast at it. How many cloaks have you ever seen like that, Miss Granger?"
So in short, to address all your points:
The Death Eaters wouldn’t be likely to suspect Harry of having a Deathly Hallow, because it’s unlikely that the Death Eaters would even know about the Deathly Hallows to begin with. Xenophilius Lovegood isn’t surprised at all when Harry, Ron, and Hermione haven’t heard of them, because very few wizards believe in their existence.
“The Deathly Hallows?’
‘That’s right,’ said Xenophilius. ‘You haven’t heard of them? I’m not surprised. Very, very few wizards believe.”
- Harry Potter and the Deathly Hallows, Chapter 21 (The Tale of the Three Brothers)
The Death Eaters also likely wouldn’t have heard about the Deathly Hallows on their own, and the Dark Lord kept his search for the Elder Wand (not the Deathly Hallows - he may not have known about them himself and wouldn’t care if he did) to himself, so they wouldn’t know from the Dark Lord’s information either.
“But to be sure, to be utterly sure, he must return to each of his hiding places, he must redouble protection around each of his Horcruxes … a job, like the quest for the Elder Wand, that he must undertake alone …”
- Harry Potter and the Deathly Hallows, Chapter 27 (The Final Hiding Place)
Therefore, it’s unlikely that the Death Eaters even knew about the Deathly Hallows to be able to suspect Harry’s invisibility cloak of being one.
It also wouldn’t seem unusual to the Death Eaters that Harry would have an invisibility cloak. Invisibility cloaks are rare, but their existence is well-known in the wizarding world - Ron is able to recognize one in his first year.
“Harry picked the shining, silvery cloth off the floor. It was strange to the touch, like water woven into material.
‘It’s an Invisibility Cloak,’ said Ron, a look of awe on his face. ‘I’m sure it is – try it on.”
- Harry Potter and the Philosopher's Stone, Chapter 12 (The Mirror of Erised)
Other wizards have owned invisibility cloaks - Moody had two.
“Found it last night when I was looking for my spare Invisibility Cloak, seeing as Podmore hasn’t had the manners to return my best one … thought people might like to see it.”
- Harry Potter and the Order of the Phoenix, Chapter 9 (The Woes of Mrs Weasley)
Invisibility cloaks are also mentioned in “Fantastic Beasts and Where to Find Them”, a very common wizarding book that every Hogwarts student has read.
“Demiguise pelts are highly valued as the hair may be spun into Invisibility Cloaks.”
- Fantastic Beasts and Where to Find Them
It wouldn’t seem unusual to the Death Eaters that Harry Potter, the heavily favored “Boy Who Lived” would have an invisibility cloak, since he’s considered important to keep hidden.
In addition, there’s nothing so obviously special about Harry’s invisibility cloak that the Death Eaters would become suspicious of it with the limited information and interaction they had with it. Though he knew of their existence, to figure out if it was a Deathly Hallow, Dumbledore borrowed it from James Potter and examined it.
“You have guessed, I know, why the Cloak was in my possession on the night your parents died. James had showed it to me just a few days previously. It explained much of his undetected wrongdoing at school! I could hardly believe what I was seeing. I asked to borrow it, to examine it. I had long since given up my dream of uniting the Hallows, but I could not resist, could not help taking a closer look … It was a Cloak the likes of which I had never seen, immensely old, perfect in every respect … and then your father died, and I had two Hallows at last, all to myself!”
- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)
Dumbledore, who knew of the Hallows, explains in his notes on “The Tale of the Three Brothers” that though the cloak described in the legend is uniquely durable, true invisibility cloaks do exist.
“True invisibility cloaks, though rare, exist in this world of ours; however, the story makes it clear that Deaths Cloak is of a uniquely durable nature.22”
- The Tales of Beedle the Bard
With only the knowledge that Harry has an invisibility cloak and very limited interaction with it, and without knowing of the Deathly Hallows, the Death Eaters would have no reason to suspect that Harry’s own is any particularly unique type of invisibility cloak.
The Death Eaters didn't go after the Elder Wand, Voldemort did. There is a distinction - Voldemort never told anyone that he was looking for the wand, even Snape, who he believed was his most loyal supporter.
Nobody really knew about the Hallows, and the credibility of their existence is brought into more doubt by Xenophilius Lovegood believing in them - as Ron says, they were regarded by most wizards as a fable in order to make people behave themselves.
The true age of the cloak was not public knowledge. Neither was its resistance to spells.
Without knowing either of those facts, it might as well be any other invisibility cloak. They’re rare, but we hear of several others during the course of the books, such as the one Barty Crouch Jr. said he was hidden under in book 4.
In addition, “very, very few wizards believe” in the Deathly Hallows, as Xenophilius says in book 7.
For Death Eaters, the far better prize was Harry himself, since Voldemort was interested in him (not his invisibility cloak).
The current answers are correct and pretty much sum it up, but I think there is another interesting answer to be had.
Remember the moral of the story of the Deathly Hallows!
To quote Tolkien, the race of men desires power above all things. The Invisibility Cloak was chosen only by the wisest of the three brothers. For most people, the Elder Wand would have been the most desirable prize to be had, given the chance to choose between the three Deathly Hallows.
Yes, it is true that most people did not believe the Deathly Hallows were real, so it is a moot point. But if they had, I think it makes sense that the Elder Wand would have been greatly sought after while the Invisibility Cloak neglected.
