What feature of Earth would be most likely to attract the interest of a curious alien intelligence searching the Milky Way marking the Earth as an unusual world and how close would they have to be to detect it?
Assume the alien intelligence has the capabilities of today's humanity in detection technology.
Life
I'm sure that there are planets with liquid water out there, planets with a human breathable atmosphere out there - regardless of all that - Life would likely be the most interesting thing.
Who are we? How do we act? What do we do? How do we live? How do we die?
There are questions enough about just the Human species that we, ourselves, don't know yet - and so any alien species with a shred of curiosity is going to be interested.
In terms of detection - I'm not sure the distance - but the thing that would most likely cause them to investigate would be non-natural EM spectrum transmissions. Radio waves or similar.
Something with a Mathematical basis.
We can detect EM/F emissions from pretty far away in the Galaxy, so it's possible they could too - the question is isolating our transmissions from naturally occuring noise.
The sizes of the Sun and the Moon in the sky are almost equal, making solar eplipses possible and especially spectacular. There is no particular reason why this should be the case - we just got remarkably lucky [link]. (I vaguely remember reading a sci-fi novel where that was a plot point, but I can't recall which one that was.)
Assume the alien intelligence has the capabilities of today's humanity in detection technology.
This means that they hardly can detect us. For comparison, we just started to find exoplanets roughly 30 years ago. And most of we found were gas giants. Very few exoplanets Earth-sized or smaller are known. And almost all of the know exoplanets were detected by indirect methods like transit, radial velocity, astrometry, transit time variation or microlensing.
Very few of them were photographed, and all those that were are gas giants presented no more than a small pixelated blob in an image already severely distorted due to the necessity of filtering out the obfuscating star luminosity.
Even looking at Proxima Centauri, which is the neighbor next door, the first indications of a exoplanet there came only in 2013 with the confirmation only in 2016. The second and third planets were detected only in 2019 and confirmed only in 2020 and 2022.
So, an alien intelligence would probably don't see us at all with our current level of technology. However, after gaining insight from their first observations, what they would be looking for is exactly what we are looking for, a planet that seems to be capable of life.
And the best way to look for planets that have life is to search for planets that are rocky, have the correct temperature range (i.e. Goldilocks planet, not too hot nor too cold) and hint away the presence of water. We know that planets like this can harbor life because Earth is one of those. Gas giants in a Goldilocks position are also candidates for having moons with some life. Surely there could be life out there outside the Goldilocks zone, and even Mars, Europa, Titan and Enceladus here around the sun are considered as possibly harboring microscopic primitive life. But what we knows for sure that works are the Earth-twins.
How close they would have to come? With the current technology, except if they are extremely luck to spot a Earth-transit, too close to be viable or realistic, but the technology is increasing very fast.
In fact, they are likely to detect Jupiter and Saturn first. Then, only after much more detailed studies with a lot of observations and luck, they could see Earth. But they would also likely be investigating millions of stars at once and having limited resources for studying them. So, detecting Jupiter and Saturn might be like "meh" for a few decades until the "meh" turns to be a "wow".
Free Oxygen in the atmosphere
Oxygen is a pretty reactive chemical, and free oxygen in an atmosphere would tend to get bound up with other stuff -- for example, it would bond with iron to form "rust" as is the case on Mars. In order for our atmosphere to have so much free oxygen, something has to be continually producing or freeing up new oxygen. That's a clue that there's life (specifically, plant life) on this planet.
Other answers have covered various aspects of "Earth has life." But what about the big ol' rock herself?
It turns out that Earth is the only planet (that we know of) to have plate tectonics. This would be an irresistible curiosity for alien geologists, as subduction zones would contain naturally formed rocks that would be entirely unfamiliar to them.
We were able to observe the crust structure on Venus with orbital radar to confirm that it lacks subduction zones entirely. But it's much easier to detect a presence than an absence - sufficiently powerful telescopes should be able to show aliens doing a fly-by that there is something funky going on with Earth to give it all these jagged mountains, and a closer look would be required.
This was going to be a response to the answer from @Lyall Stewart in Agreement with @Nosajimiki. but became a bit long. In the galactic population, are predator or prey species the type that generally evolves to sentience? There is a genera(?) of Sci-Fi where humans are referred to as "Deathworlders" because they are the rare exception to sapiants since usually prey species are the ones who commonly evolve, sometime even more rare, Omnivores.
If the earth was much larger, achieving the tech necessary to counteract 9.81 m/s² acceleration to leave its gravity well is approaching impossible with chemical rockets. Thus Just by being born on a ""High G planet" Humans may be among the most physically strong and durable spacefaring sapiants.
We breath one of the most reactive / corrosive elements on the periodic table, Oxygen.
50 to 60 milligrams of nicotine is deadly to a 150lb adult. We consume this substance as regularly and recreationally.
98.6 F (37 C) resting human body temp, is enough to cause serious burns to some species if physically touched.
In less than 8 generations from when humans developed industrialization they were able to destroy the nearly all life on their home planet.
Population expansion ratios are among the highest of native species above 50kg.
Im sure hundreds more examples like these can be suggested for a universe where humans/earth life is exceptional in the universe.
I think you are asking what might cause aliens on distant worlds take an interest in Earth, of all planets.
The answer is certainly an atmospheric composition that betrays the existence of life. The atmosphere of exoplanets is examined by spectral analysis, which can be done with the proper telescopes. Humans are at the cusp of deploying telescopes capable enough of doing that routinely, like the JWST. One answer mentions free oxygen; but anything off-balance is a sign for life. This article appears to give a good overview.
If, by contrast, you ask what aliens might find interesting once they are here (or what they may find interesting in alien civilizations in general), a number of ideas have been explored in the past:
In Charles Stross' Singularity Sky, an entity called The Festival drops cell phones from orbit and promises through them: "Entertain us and we will give you anything you want." They are a nomadic intelligence absorbing information and stories.
I think information, in particular cultural information like songs, texts and images, are the most likely items of value Earth could offer spacefaring civilizations. Additionally, actual material artifacts may have value as well.
Robe Reid imagines, in his novel Year Zero, that by a twist of fate humans are the only race in the universe which is able to produce good music. Terranean music makes aliens ecstatic. It is cult. They name their heavy metals after rock bands: vanhalium, slayerium, megadeathium.
"What about bonjovium?" I asked. I've always had a weakness for "You Give Love a Bad Name".
"Of course it exists. But bonjovium is certainly not a heavy metal by our standards." Özzy sniffed. "It has an atomic number of just fifty. You call it 'tin'."
The copyright for the Terranean trove of music is of immeasurable value.
In the DUST short film Final Offer, featuring Anna Hopkins who also appeared in the Expanse, the aliens are interested in the water on Earth. Given that part of the water on Earth probably comes from asteroids, it seems easier to harvest those than to siphon it up from the bottom of this gravity well. (But the short is still very well done and entertaining.)
We don't know if life is anywhere else in the galaxy, however if there is, there is an argument that such life would usually likely be microbial, non-technological life.
Life has been present on Earth for a long time (over 3 billion years), yet only in the last few dozen years have computers, and artificial life been possible.
So it is conceivable that Artificial Intelligence may be the unique 'turning point' alien civilisations may be looking for to encounter, or satiate curiosity or to conduct reconnaissance.
Such Artificial Intelligence could manifest itself in terms of transmitting radio, communication via light or some other evidence that causes alien civilisations to now take Earth seriously. Communication with our new AI could be at an amazingly fast, iterative, and improved rate, causing essentially the birth of a new era, influenced by / influencing the Alien Intelligence (which, by the way, may be an AI itself).
Plus it is no secret that AI may inevitably become a dominant force on this planet, and can easily also become a dominant force on all planets in the galaxy, so best to now have a look.
Life on Earth tends towards the aggressive side, where it will destroy rather than assimilate competition. This can be useful if used correctly.
Wood is rarer than gold on a cosmic scale.
Interstellar alien loggers use an alien James Webb Space Telescope and alien ramjets to find, and pillage, oxygen rich worlds. Evolution favours tree analogues on earth-like gravity.
Alien culture mines alien asteroid 16-Psyche; alien metal prices plummet and demand for alien wood increases.
Oxygen would attract their interest. Wood would keep it.
Water
One feature of Earth that is unique among extraterrestrial planets is its abundance of water. The Blue Planet is covered in the stuff.
