Weirdly enough, Araya must, in fact, be referring to the fact that "Lio" is one English letter away from "lion". This has to be the case because "lion" in Japanese is 獅子 shishi, which bears no resemblance whatsoever to Shirazumi's name.
Of course, Araya should be fluent in English, having lived at the Clock Tower in London for some years, so it's not that weird. And Nasu (the author) clearly knows enough English to give us silly phrases like "blue blue glass moon" and "heavens feel", so I wouldn't put it past him to put a touch of English wordplay into Kara no Kyoukai
(Some guy on chiebukuro claims that it's actually that リオ "Lio" is one Japanese character away from レオ "Leo" as in the zodiac lion, but I think that guy's claim is dumb.)
(Side note about Japanese: English "lion" is pronounced ライオン raion, so you can't actually get Japanese raion from "Lio" with a single character in either English or Japanese. But you can get English "lion", which is why I say that that's what Araya must've been going for.)