Terra, meaning Earth in latin. Terran = of Earth. Unsure why this has been such a staple of sci fi to use the latin name. I guess it's more classical than Earthican (lol).
Anyway, perhaps As Terra is the only planet of our system to have sentient life (as far as we and most sci fi authors these days acknowledge), especially in the star trek universe it seems to be practice to name the planetary system after the home planet of the dominant species, e.g Romulan system, Cardassian system etc. Given some of the first human ships and deep space probes had latin on their insignia or badges, or in the messages they contained perhaps the first recognisable name for the species some aliens encountered was Terran and it stuck.
As for us humans - if we had remained a geocentric society as opposed to a heliocentric, we would perhaps be calling it the Terran system ourselves rather than the Solar or Sol system.
For the common gamer and Trek fan, it would have been better to rephrase the line and instruct you to go to "Planet Mars in the Sol system".
"My frozen dairy-based confectionery attracts all the males of the species to the facilities. They all agree on it's superiority. Indeed, it is superior to yours. I could teach you the finer details but that would require monetary recompense on your part." -The Milkshake Song: Vulcan Edition