So, I finally decided to Google it.
The 「ダン」, as it turns out, is 「段」, which can mean tier(ed). This is most likely because cardboard is corrugated, and if you were to look at a cross-section, you would see two or more little layers kind of connected by ridges. Fair enough.
As for the 「ボール」part, which looks like "ball," I came across one explanation offered at Yahoo Answers-type websites, and one that Wikipedia kind of hinted at that I am going to proffer.
1) Whoever coined the Japanese word for "cardboard" misheard or had trouble pronouncing "board" and said 「ボール」(ball) instead. It caught on and spread, and voila. This was an answer I came across multiple times across the webiverse.
2) 「ボール」is actually short for 「ボールド」. I think this is plausible since so many words in Japanese can be and often are shortened. Wikipedia didn't explicity say this, but it did say the origin of 「段ボール」's 「ボール」 is the English word "board."
So there you have it. Man, now I wish "cardball" were an actual Japanese game.