In the Star Trek franchise, do the letters in the title "Borg" represent anything other than an abbreviation of the term "Cyborg"? If it is an acronym, what does "Borg" stand for?
No, it has no extraneous meanings. It is a contraction of the word "cyborg" as you have denoted.
- The most curious part is why the Universal Translator would choose the word "Borg" in the first place. It implies the Borg and Federation Galactic have common shared words, or perhaps the Borg created a word that would have the proper connotation from the Federation database.
- Language translations do not usually support contractions of words from language to language.
As Iszi's request (and since at least one other person agrees), I'm pulling my comments into this answer and expanding on them to get a possible answer for why the name "Borg" could actually be a contraction for "cyborg" in-universe.
This is the first time we hear the word "Borg", in Q Who:
A Borg cube appears in front of the Enterprise
Guinan: I'm here, Captain. Viewscreen's activated, I have the other ship.
Picard: You are familiar with this life-form?
Guinan: Yes. My people encountered them a century ago. They destroyed our cities, scattered my people throughout the galaxy. They're called the Borg. Protect yourself, Captain, or they'll destroy you.
These are the last lines the Borg have in The Best of Both Worlds before Picard is assimilated and Locutus addresses the Enterprise:
Picard: Impossible. My culture is based on freedom and self-determination.
Borg: Freedom is irrelevant. Self-determination is irrelevant. You must comply.
Picard: We would rather die.
Borg: Death is irrelevant. Your archaic cultures are authority driven. To facilitate our introduction into your societies, it has been decided that a human voice will speak for us in all communications. You have been chosen to be that voice.
Then Locutus hails the Enterprise:
Locutus: I am Locutus of Borg. Resistance is futile. Your life as it has been is over. From this time forward, you will service us.
The name "Borg" was not spoken by any of the Borg between those two points, giving the impression that they started using the name "Borg" when in contact with the Federation because we already knew that name. Like Locutus originally being the human Picard, it acted as a common reference point between the two cultures.
Now, as for why we knew that name and not another - Guinan never said that they call themselves the Borg, but that's what others call them. At first it sounds as though it couldn't have originated from "cyborg" (coined on Earth in 1960), but in Time's Arrow, we learned that at least Guinan was on Earth in the 1800s. The Borg were not mentioned in Time's Arrow, but her people could easily have come and gone in the intervening centuries.
Cybernetic enhancements in the Star Trek universe seem to be relatively rare - only the Borg, Bynar, and maybe the Vidiians (Denara Pel in particular) actually practice it regularly, so a word equivalent to "cyborg" may not be very common for other languages in the galaxy. I mean, we got a look into Cardassian fiction in DS9 2x22, The Wire, which highlights how different that particular society is to ours - it gives a slight impression that the Cardassians were unlikely to have developed a parallel word in their culture. (However, it is the opinions of only one Cardassian, so their literature could be as varied as our own; it's hard to tell)
So when the Borg attacked Guinan's planet about century prior (which would be somewhere around 2250 - 2280, roughly the era of TOS), they may have only had the imported Earth word "cyborg" as a reference point. Hence, Borg.
In light of how the Star Trek writers love to immortalize 20 Century, and older, scientific figures in their movies, could they have been named after Anita Borg, Computer Scientist? Who, in 1987, Borg founded Systers, the first email network for women in technology. After all the leader of the Borg is a Hive Queen . . .