In the Zohar quoted in yesterday's (first day of parashath Terumah) Hok L'Yisrael, I found an interesting comment:
When HaShem [the Shechinah] arrives at the Beit Knesset in the morning, and no one has arrived yet, He gets enraged and demands His honor! On the other hand, if there is even one person in the synagogue, so He says (something like) "This one will be for me." And HaShem develops a special relationship with this person, apportioning him the title of Tzaddik.
The aramaic word used there meaning "for me" or "to me" is none other than Gabai (גבאי) !
So, perhaps the one who looks after and opens the Beit Knesset is called the Gabai, because HaShem Himself comes and proclaims, this Tzaddik is "for me!" (Gabai)
[as a curious aside, I only recently discovered that my family is a scion of the Gabai family of Baghdad; my grandfather told us that we were of the Kabir clan back in Baghdad, and I recently found out that the [Al] Kabir clan/family was the name by which (at least parts of) the Gabai family was known. Which makes me very indirectly related to the Ben Ish Hai. But, I figured I'd mention this here, in the interests of full disclosure]