Satake (in his PNAS paper where V-manifolds are introduced and in his Journal of the Mathematical Society of Japan paper where Gauss-Bonnet theorem for V-manifolds is proven) never explains the origin of the name. If I were to guess, I, as temp, would say "V" stands for "virtual" (since, for instance, in topology and group theory the word "virtual" means "up to a finite cover" or "up to a finite-index subgroup", and orbifolds/V-manifolds are "locally" manifolds up to a finite branched cover). But my guess is as good as yours and we are left with a mystery. Maybe, as Igor says, "V" stands for a "cone singularity" because of the shape of the letter V (could have been a $\Lambda$-manifold just as well). Or, maybe "V" does not really stand for anything, like R.H.Bing's initials. (According to wikipedia, once R.H.Bing was applying for a visa and was requested not to use initials. He explained that his name was really "R-only H-only Bing", and ended up receiving a visa made out to "Ronly Honly Bing".)