name & rank
ARE YOU SURE IT'S NOT RIVAILLE?
Many internet arguments have been waged over the spelling of Levi's name. In scanlations, it is sometimes romanized as 'Rivaille.' This was a valid guess, as there was really no indication of how his name was intended to be written in English. In Japanese, his name is written リヴァイ (rivai). The spelling used in the official Kodansha manga is Levi.
In his blog, Isayama commented that he got the name from Jesus Camp, a 2006 documentary about an evangelical Christian camp for kids—one of the kids is named Levi. It's a pretty terrifying movie. That said, there doesn't seem to be anything especially meaningful about the name choice; he just used it because he liked it.
Since we already know where the name came from, I don't really put much stock into its "meaning," but for the sake of completion, according to Behind The Name 'Levi' is a Hebrew name that means "attached" (or "joined"). It is also a biblical name, though I don't think that's necessarily significant, either.
RANK
Levi is usually referred to as 兵長 (heichou). Because the translation of 'heichou' is 'lance corporal,' this was commonly thought to be his rank. HOWEVER, this is incorrect! It is actually a shortening of 兵士長 (heishichou), which would be more properly translated as "captain"—in fact, it is the official Kodansha translation of his rank.
The more you know.