The riddle was never revealed in full, so we have to work backwards from the answer
It is established that classical Latin is derived from Ancient, so they share a number of words. So figuring out the riddle will first involve figuring out the meaning of "Contagia".
It's quite likely that the word is different in Ancient than it is in Latin. As Valorum pointed out in the comments, the earlier riddles had the answers "ventio" (ventus, meaning wind) and "clavio" (clavis, meaning key)*, which were nouns.
The word as-is is one of several plural conjugations for the Latin word "contagio", meaning "touch" or "contact". This doesn't seem to fit with the riddle though, as it's not clear how touches can be battle-scarred.
However, the Oxford Latin Dictionary translates the word as "disease" (in reference to many diseases being transmitted through physical contact). This reflects the modern word "contagion", and is pretty close to spelling of the riddle answer "contagia".
Therefore, I think the answer is "disease".
This seems to fit with the riddle:
Battle-scarred in times of strife
Diseases are often rampant during wartime due to injuries, the transport of troops from foreign regions, and the logistical challenges of fighting.
Diseases can also become resistant to treatments; there are often concerns of bacteria or viruses mutating to become immune to treatments.
The full riddle itself was never revealed. Perhaps the full statement was something along the lines of "Resistant to being wiped out" or something along those lines, referring to the tenacity of diseases.
But how did they know the answer?
Note that Adria, the leader of the Ori, was the one who provided the answer. How did the Ori begin their war on our galaxy? With the vast spreading of disease, which we saw quite prominently on Earth during the Season 9 two-parter "The Fourth Horseman". Since the riddle said it was something that is battle-scarred that is resistant, that was probably something prominent on her mind.
* Valorum speculated that it could be a variation on the verb "contego" (note that the "a" changes to an "e"), meaning conceal, protect, clothe, bury, or entomb. Valorum speculated that since the other two answers were nouns, this might be "protection", or more concretely, "a shield". However, Latin doesn't have a noun form of this word (i.e. it is only used as a verb), so I disagree and think that "contagio" is a more likely choice.