In the original Japanese game, Contra is meant to be the name of the fighting force made up of our two heroes: Bill "Mag Dog" Rizer and Lance "Scorpian" Bean. Furthermore, Shattered Soldier defines a Contra as "a title awarded to a superior soldier possessing almost super human drive and bility, while excelling in guerilla tactics." But as it turns out, there are many more layers behind this title.
Guerilla groups in Nicaragua, skilled in jungle warfare, were known as "contras". Given that the original game starts off in a jungle, this is definitely not coincidental. They probably got their name from Spanish, where "contra" means "against". It's also a Latin rootword that means "opposed to" - looks at words like "contrary" and "contrast". This meaning can be applied to acts of agression and wars.
The way "Contra" is written in Japanese is somewhat interesting. It is composed of three kanji (Chinese characters that represent an ideal) to create one made-up word. The first one, pronounced "kon", means "spirit"; the second, pronounced "to", means "battle" and the third, pronounced "ra", means some sort of cloth. When sounded out together, the result is "kontora", which is how "Contra" is pronounced in Japanese. This is known as a "gikun", where the kanji is used for phoentic pronunciation over actual meaning. However, if you want to look at the direct translation anyway, you get "Battle Spirit Cloth". This seems a little esoteric, but it could possibly be applied to the Rambo-esque headband worn by the heros in the games' earlier installments.
Also of interest is how the title of the American NES game was shortened to "Super C". The most logical explanation is that Konami perhaps wanted to avoid any connection to the Iran-Contra scandal of the 80s, where American supplied various military supplies to Iranian terrorists. Super C came out in 1990 and the Iran-Contra affair was pretty much over at the end of President Ronald Reagan's term in 1988, but perhaps Konami was simply playing it safe.
"Contra" is also a kind of folk dancing, as it turns out. Uh...yeah.