A device is recovered from an impact site in the desert. This device contains bizarre, but not inconceivable, components and a simple interface. The interface is a black and white touch screen with a display showing many versions of "Press any key to continue. " scrolling by on the screen in every earthly language. After being pressed the device insists it is of alien origin and represents the first attempt to make contact between its creators and the people of Earth. The scientists working on this are skeptical and demand proof that the device is not just an advanced piece of hardware designed by another government to distract them from their work. Without other worldly hardware or materials how can the device convince the scientists it is alien via a simple text interface? For reference this takes place in the modern day.
The easiest way to identify itself as of alien origin would be to provide a large amount of scientific information that is not available on Earth but which could be tested immediately and or in the coming years and decades.
For example, describing the theory of everything that physics has been searching for as well as a test that could be conducted to confirm that theory. It could also describe how to conduct further tests that can’t yet be performed but are almost within our grasp (you need this part engineered to within one nanometer...).
Other possibilities include revealing astronomical facts that we are not yet aware of: The cavern of methane geysers on Titan, together with its exact location, the only outcrop of Thorium on Mars that would be classified as a rich ore on Earth, the exact orbital parameters of a range of bodies in the outer solar system that we have not discovered yet like the bizarre distant group of three small bodies that orbit each other. Or the neutron star that we can’t see because it is obscured but might be detected by a space probe (https://archive.org/details/dragonsegg00robe/page/286/mode/2up)
With a careful choice and many examples it should be possible to give over whelming evidence. This could be confirmed today in some cases since they told us where to look or how to conduct the experiments and in the coming years when our technology has progressed sufficiently far.
Show things only aliens might know
The aliens are bound to have a bigger understanding of the universe. They somehow cracked every language on Earth, so further knowledge isn't a stretch. In that vein you can go much further. As they know every modern language it is very likely they know the state of modern technology. With this they can direct scientists to fringe discoveries that aren't known yet.
For example, current scientists are very well aware what research is being conducted in space. These are limited resources that can't just have other discoveries. If you give them an accurate result and direct them to this new discovery, they will know that at least something with more knowledge than them is interacting with them. It might also predict some things accurately that we don't currently understand. There is a weird phenomenon around a certain star that is difficult to predict and explain currently (which some suggest is an alien structure). It might predict this accurately.
This doesn't need to be limited to space though. The aliens clearly do have some technological advantage with knowing all languages. That means they can divulge some scientific information and theories to further our understanding. By doing this they can be reasonably certain it isn't a hoax, as well as making any doubters still not dismissive of any further information.
It does make this a Pandora's box for the scientists. The question is if they should open it. I remember one telling of the Greek story where it was opened twice. The first time it brought something good, but it was warned to be never opened again. Pandora was curious and did so anyway, hoping for more good. All disease and death escaped into the world, making it a much worse place.
Following an unknown possibly alien intelligence their directions can be very dangerous. Even if it's only used for communication it can be hazardous thanks to the aliens manipulating, or humans just being humans and going crazy. Regardless of the possibility of a hoax, scientists would be careful.
It should be as simple as eliminating any possible scam motivation. This isn't easy though, because humans are inclined to create hoaxes for the sake of hoaxing.
However, when that happens, it is rarely more than a handful of people in on it. Most bigfoot/cryptid sightings are 1-3 people, and the crop circle folks had up to half a dozen. This device might only require a handful of people, but what of an analysis that confirms that no fewer than dozens/hundreds of humans would be needed to create it?
This eliminates the possibility of a hoax, I think. Relegating it to a project only nation-states could hope to complete.
Not just any, either. Only those with electronics fabs could do so without farming it out to third parties and prematurely exposing it. And what of the components themselves? Sure, they may be very similar to our electronics, but if this thing is using -8.2V and is some weird CMOS technology, at some absurdly wrong process node size... some of this stuff just becomes impossible to adequately fake. You realize that fakers couldn't make only one of these, because you often have to make thousands or even millions before you get good enough at them to put it into a retail-ready product, right?
Finally, you have to start to wonder just what the grift actually is. Even the most elaborate fakery is easily revealed by the fact that, if false, some other party has something to gain by it.
What would Taiwan or South Korea or China have to gain? Not much. These aren't nuclear weapons scientists that are being distracted, these aren't aircraft carrier logistics folks, they aren't economists who will be distracted. Just electronics nerds.
More so, they all have something to lose if they pull this stunt. They've given a valuable piece of hardware to their enemy who will then easily deduce some of their most top secret technology secrets just by examining the device. "Look at this Joe, China can crank out functioning prototypes of supposedly alien-technology as one-offs with a new CMOS tech at a really funky base voltage like you or I could scribble on a post-it-note, and then crash it into a crater in the Arizona desert!".
No, if they could fake this, they wouldn't do it.
The device itself does not have to prove itself, it has proven itself merely by existing.
Show future knowledge.
If the device has ftl communication between it and the creators, then it by nature is a time travel device. If there are stellar events closer to them than you, they can report what the light from the stars does before you do, and relay this information.
Do computational tasks faster than them.
The earth scientists could also send them a problem which takes a long time to solve, but which can be easily verified, like how a protein folded, or the factors of very large numbers, used in crytography.
A much more advanced civilization presumably has much more powerful computers and can solve such problems much faster.
This could be used by scientists to earn vast amounts of money.
Provide precise measurements of things the LHC and other particle accelerators are measuring.
They presumably have better particle science than us. They could give predictions about the particle physics of particular reasons that no one has scanned, which could be verified later by the collider.
All elements on earth have a particular atomic signature. Ratios of elemental isotopes contained in a sample from the devices construction. For instance if you made a knife on earth, and made and "identical" knife on mars. You could analyze the ratio of iron among all of the other present isotopes to identify not only which knife was Martian, but where on earth the earth knife iron was mined. This is a widely used research method for things like identifying meteorites origin. Origin of gold or other precious metals in criminal investigations or commodity audits. archeological forensic studies. Thus if they tested the device and it were show to have materials that match the isotope ratio of aluminum mined from a know source on earth, one would assume it would likely be a hoax.
I know that the question specifies that it shouldn't be other worldly hardware, but many of the answers talk about faster computation and similar things. As this fundamentally needs better hardware I'll assume that tech that we know is possible, but barely can't make is ok.
So why does it almost certainly need to be a physical thing? Very simple. Providing advanced mathematical or physics equations is not a proof. A until now unnoticed genius could have easily created them. They are not fundamentally impossible to be of human origin. Similar with any space data. Anything that we can verify from earth could have also initially been derived from earth. As such if it should be some sort of space data then it would need to be some kind of data about or solar system that needs a space probe at a specific point to measure something. As a result it would take years to confirm this, as space probe for interplanetar travel take very long to build. Occam's razor may be a good starting point, but it is not a proof. It is based on logic and can only deduce what is likely and unlikely, but just because something is unlikely doesn't mean it's impossible.
The following part will most likely sound like gibberish if you don't have a basic understanding of photolithography, but if by the end it sounds vaguely like a good solution I'd recommend to look a bit into the topic. If I also explained all of the necessary background knowledge then I'd still be here tomorrow. Plus there are plenty of great sources that explain the basics.
As such that only really leaves the hardware. And this is where it gets interesting. On earth there is currently only one company (ASML) that is able to produce the machines for UV/EUV lithography. This has been the case for over two decades now and no state actor has been able to reproduce these machines (china has tried very hard). During photolithography a special pattern needs to be placed in front of the laser, such that due to refraction and reflection in the material, only the intended part is actually being worked on. These advanced patterns are increasingly difficult to simulate, but not impossible. More complex patterns allow for smaller features on the wafers.
As such the computer could have the starting parameters for the given simulation, so that we can compute it ourselves to prove that it actually should result in that shape. Then the alien device could have these tiny features etched on its processor, as proof. This would work, as photolithography is one of those fields in science where we exactly know how things will look in future, but we are currently still in the process of creating equipment precise enough to create these things. In addition we also very clearly know the best technology that humans currently have and can claim with almost 100% certainty that there is no way for this to be of human origin.