Some years ago, I had to make a costume for an International Space University party and thought that I would go as a space archaeologist. I found a yellow fluoro work vest and used a black texta (sharpie) to draw a logo of the fictional Institute of Space Archaeology. Then I made a name tag, using an old plastic sleeve from a conference, and invented the character Tasmania Smith, Director of the lunar office of the Institute of Space Archaeology (ISA). At the bottom of the name tag, I wrote Tasmania Smith's email address:
tsmith@isa.luna
This is possibly the first ever specific lunar domain name.
It was just to add a little authenticity to the character, but when I came across the photo again recently, it made me wonder if provision had been made yet for off-Earth domains. I went looking for academic or industry literature on this topic. I looked in Google Scholar, the NASA Technical Report Server, and Arxiv, and generally puddled around on the internet. As far as I can tell, there is no literature on how domain names would work in space.
Certainly at this stage astronauts just use the email addresses of their countries or employers - there is no specific International Space Station address, for example - crew use nasa.gov, or esa or Russia or whatever. Interplanetary internet connects spacecraft of all kinds with Earth and each other.
But it seems there is no Domain Name System (DNS) for space yet. The Interational Standardization Organiztion is responsible for domain names on Earth. ISO 3166-1 alpha 2 establishes the two-letter country codes. So I suppose they could do the space codes too.
Eventually, each planet or asteroid might have its own suffix. There doesn't seem to be any commonly used abbreviations for the planets, even by the International Astronomical Union. You could make an abbreviation out of the Latin or Greek names, but the trick is to make sure that they aren't duplicating national ones. And yes, celestial body names are an expression of European hegemony and colonialism, as are many country names, but they are commonly understood at least.
So I went though the list of two-letter country-code top-level domains to find those that might be unused and available for planets. A harder task than it seems! Many of the most obvious two-letter codes already belong to a country. What's left isn't always the most intuitive thing to call a planet. However, there are so few planets compared to countries that perhaps a whole word or abbreviation would work. For the Moon and Mars, all the obvious two-letter codes are already in use.
So we might have:
Mercury - He, short for Hermes, the Greek equivalent of Mercury
Venus - Ap, short for Aphrodite
Earth - Ea, or Te for Tellus, or Gaia as a whole word
Moon - Luna
Mars - Mars
Jupiter - Jv, short for Jove
Saturn - Cu, short for Cronus, the Greek equivalent of Saturn
Neptune - Po, short for Poseidon
Uranus - Ur, not taken for any country yet!
Pluto - Pu. the closest available abbreviation
Sun - Sun
For Moons other than Earth's, the relevant planet's domain can be used. Just to be precise, we might define the spatial extent of the domain as the planet's Hill Sphere. It's likely that we wouldn't need anything beyond Mars in any case.
There might need to be special ones for spacecraft moving between different planets. I'm not sure why they'd need one, but perhaps they have human or AI inhabitants. A mobile domain might present its own issues to resolve.
In the early years of lunar habitation, crew or inhabitants will still likely use national or agency domain names. The Outer Space Treaty makes nations responsible for the activities of their citizens in space. The point at which you'll need a lunar domain name is when there is lunar government separate to Earth. One of the first acts of a lunar rebellion might be to assert independence by declaring a new domain with its own code. This may also imply the presence of local servers.
A private corporation providing a domain server might get to apply its code to a planet or moon. So keep an eye on what the space billionaires are doing - this might be a step is asserting political dominance.
Maybe there will also be the equivalent of gmail or yahoo in space. Maybe there will be open source or citizen higher level domains.
Don't ask me how this works in practice, I'm not sure I understand the technical aspects!