Besides Latin, of course. It seems to be very common to refer to the (typically colonized) moon as "Luna" in sci-fi works. Did authors independently decide to use the Latin name? Or did someone originate this, and future usages were a reference/tribute? Why do future humans use a name that current humans generally don't?
Primarily, 'moon' is too general. Luna refers to Earth's moon, not other moons, of other planets. People outside SF&F have no reason to think about any other moons, for the most part, so saying 'The Moon' is distinct, and understood. In a setting where you are talking about other planets and THEIR moons, saying 'The Moon' is meaningless, particularly between people from different planets.
The reason why people in a science-fiction future would call the moon Luna is explained in Seeds' answer. If, like most people today, you only know about one moon, you can just call it the moon. If there are lots of moons that play a part in your life—you read about them in the papers, visit them on vacations, go to work on them, etc.—and if all the other moons have proper names like Phobos, Callisto, Titan, etc., then Earth's moon needs a proper name too, and Luna is a natural choice.
But you also seem to be asking about the origin of calling the moon Luna in science fiction. According to the Historical Dictionary of Science Fiction, it seems to have started with this line from Raymond Z. Gallun's novelette "The Lunar Chrysalis" in Amazing Stories, September 1931, p. 528, column 2 (available at the Internet Archive):
I never regretted my decision to be one of the first men to visit Luna.
(...) use a name that current humans generally don't?
Wow, quite a claim there!
- 427 million of native Spanish speakers call our moon "Luna".
- 160 million of native Russian speakers call our moon "Луна".
- 65 million of native Italian speakers call our moon "Luna".
- 24 million of native Romanian speakers call our moon "Lună".
- 9 million of native Bulgarian speakers call our moon "Луна".
So we have 685+ million people that say, exactly, "Luna" when talking about our moon.
- 230 million of native Portuguese speakers call our moon "Lua".
- 80 million of native French speakers call our moon "Lune".
- 4 million of native Catalan speakers call our moon "Lluna".
- 2 million of native Galician speakers call our moon "Lúa".
A total of about 1 billion people closer to "Luna" than to "Moon".
How does "Moon" compare?
- 340 million native English speakers saying "Moon"
- 40 million native Hausa speakers saying "Moon"
- 28 million native Filipino speakers saying "Moon"
- 17 million native Somali speakers saying "Moon"
- 2.5 million native Slovenian speakers saying "Moon"
So that's 425+ million people saying "Moon" today.
- 95 million native German speakers saying "Mond"
- 23 native Dutch speakers saying "Maan"
- 10 million native Swedish speakers saying "Måne"
- 7 million native Afrikaans speakers saying "Maan"
- 5.5 million native Danish speakers saying "Månen"
- 5 million native Norwegian speakers saying "Månen"
A total of 570 million people that may be closer to saying "Moon" than "Luna" (I say maybe because e.g. "Månen" is closer to "money" than to "moon", but I included it anyway).
If we include second languages, the numbers are about 1.3 billion people for "Moon" and 1.5 billion people for "Luna".
So it turns out that humans do use "Luna" as the name for our moon. More than Chinese "月亮", more than English "Moon". This and the "clarity" argument given in the other answers may be the main reasons for some future Earth using "Luna" as the name for our moon.