It's a classic story of someone having to pick a name quickly, which then gets established long before anyone who cares about branding is aware of its existence.

The original service didn't even have a name, only a description, and it was amusingly hosted at xxx.lanl.gov. But LANL wasn't really interested in it, and the founder eventually left for Cornell. At that point, the service needed a domain name, but archive.org was already taken.

And besides, the name has Ancient Greek influences. A similar Latinate term might be something like "archive".

I thought the X was an allusion to LaTeX.

Interesting, thanks for the context! Makes it more understandable as a choice.