Why does Thor entrust The Collector with the Aether/Reality Stone?

Hiding in plain sight

The Collector a.k.a Taneleer Tivan can be seen in MCU and he is actually a much more powerful being originally in comics than portrayed in the MCU

"I present to you, Taneleer Tivan, the Collector!" ?Carina (in Guardians of the Galaxy)

MCU wikia explains about collector as follows,

Taneleer Tivan, also known as the Collector, is the keeper of the largest collection of interstellar fauna, relics and species in the galaxy, operating from the Knowhere port installation. Through his dealings with Asgardians and the Guardians of the Galaxy, he was in possession of one of the Infinity Stones and was nearly able to obtain a second stone. However, this caused Thanos to attack his museum seeking the Aether.

it further explains how Tivan got the stone in the first place,

In 2013, two Asgardians, Sif and Volstagg, arrived at Tivan's museum, bringing with them the Infinity Stone known as the Aether, which had been recovered by Thor after his battle against Malekith on Earth. They gave it to him believing that it would be safe from other parties. When Tivan asked why they did not keep the Aether in their own vault on Asgard, they explained that they thought it would not be wise to keep it on Asgard since another stone, the Tesseract, was already on there; a notion that Tivan agreed with. As Sif and Volstagg left the Aether in Tivan's care, Tivan remarked to his assistant, Carina, that they now had one of the Infinity Stones in their possession, and only needed the remaining five, hinting that Tivan was already seeking the Infinity Stones. Tivan learned the location Power Stone, which was contained inside the Orb and begun working out a plan to obtain it for himself.

From an answer for a similar question in SciFi SE ,

MCU version of Tivan portrays him as just an eccentric collector so far, the comics version has a personal mission of collecting artifacts and subjects and keeping them safe so that the Universe could be repopulated. Not only does Tivan have a personal mission to keep all of these artifacts safe, he also has ten heavily guarded museum worlds, as well as a ship that remains in hyperspace almost all of the time to avoid being detected. Additionally, he has "temporal assimilator" which can be used for limited time-travel.

There was a conversation between Thor and Star-Lord in Avengers Infinity War,

Star-Lord: "If it's with the Collector, then it's not safe. Only an idiot would give that man a stone."

Thor: "Or a genius."

POV from Reddit

In MCU The Collector was a known buffoon - despite his reputable collection of things, he was a joke across the galaxy. But then the Asgardians realized that the one place nobody ever thought they'd keep the reality stone was with a well-known doofus.

It's essentially the galactic version of hiding something in plain sight, like stealing the Mona Lisa and hanging it a small town community art museum, nobody would ever think something so valuable would be in such a non-descript place with such an idiot guarding it that they wouldn't even bother to look there in the first place.

I found another POV in Reddit,

Personally, I felt the most important reason was that The Collection had no intention of using it. Odin/Loki didn't trust having two infinity stones on Asgard, so he probably wouldn't trust the Nova Corp with it either, nor the Avengers, since Earth had two already.

So, Collector can be choosen for

1. Has the ability to protect a stone,

2. Would be willing to have a stone, despite it potentially putting a bullseye on them.

3. Can be trusted not to abuse it.

The Collector wants it for its rarity and collectability, not the things he could do with it. He wasn't going to go try take over the galaxy with reality warping power. So, he was a good choice. The only question was his ability to safeguard it. Assuming no-one would find out he'd been entrusted with it, and that the Collector had probably collected a number of defence systems to protect his collection, he was probably a decent enough choice.

The Collector is not a complete idiot as Quill said. He is obsessed with collecting missing and valuable artifacts at any cost. He even offers 4 billion credits for the Power Stone which startles Rocket. Therefore, it is clear that there is no other thing he would trade an infinity stone for. So, Asgardians believed that the stone won't be transferred to anyone.

Besides this, we have seen his museum where he had strong vaults and defending means, which is why he has been capable of keeping all those artifacts safe for a long time. This could also be the reason why Asgardians trusted him.

Thor/Asgard entrusted the Collector with the Reality Stone he would neither sell it nor use it.

He would not sell it

The collector started collecting things as a hobby to maintain his own sanity. He eventually began to collect things just for the sake of it. He was collecting for many years. All the things he collected remained with him. He did not intend to sell them.

He will not use it

For thousands of years the Asgardians know him and he has not tried to dominate the world during the time. Though he says

One down. Five to go!

in the movies I suspect he told that about collecting 5 more stones and not about using them.

Safety

According to Wikipedia,

Taneleer Tivan is one of the Elders of the Universe and is close to his fellow Elder En Dwi Gast (the Grandmaster). He apparently came to self-awareness billions of years ago, on the planet Cygnus X-1. He is an enormously powerful being who wielded the Power Primordial and, though at first he took the appearance of an old human, his true form is a powerful alien

So he is powerful life form and was able to safeguard his collection of rare objects for thousands of years. He might be able to safeguard ether too.

They might / might not have thought he was able to defend against Thanos or Ronan or others. But they definitely believed that he will not use the stone which was far more important.

The end credit scene of Thor:The Dark World was made as a connection to The Guardians of the Galaxy which was to be released the next summer. The Collector was supposed to have a glimpse here for Marvel fans to know him and then see him in a bigger role in the GOTG movie.

It was obvious that Asgardians couldn't store 2 infinity stones together because they were unaware of any consequences that may happen, they decided to give it to the collector only because he is a very old entity himself, he has seen and collected NUMEROUS things over the VAST time he has lived, and they knew he knows about all the weird/unknown stuff better than anyone.

Every elder has a power primordial, which he inherits due to the power passed on from the Big-Bang ,they have become skilled and masters of their skills over billion and billions of years.

Asgardians(even Odin) knew the experience theCollector has regarding any object new or old or ancient. So, the Aether was handed over to him.However the Collector himself became greedy upon having a stone.

He himself says "one down....5 to go !"