επιουσιος seems still to be a bit of a mystery, and I frankly find it odd that neither it nor anything very much like it has yet turned up in papyri (other than as a misreading). Superficially it may look parallel to περιουσιος, which might imply analysis as επι-ουσιος(?!). But the meaning of περιουσιος in its occurrence in the Septuagint, while seemingly unique, is perfectly unexceptionable, perfectly in line with περιεῖναι denoting superiority. The same can’t be said of επιουσιος vis-à-vis επεῖναι!
I think we should be looking not to εἶναι but to ἰέναι. I’d take επιουσιος to be an adjective formed on the basis of the very common ἡ επιουσα (ημερα), see LSJ ἔπειμι (B) II.1. Νo other explanation seems at all plausible to me. Then the prayer is for bread for the coming day (as distinct from bread for subsequent days, cf. Matt.6.34 μη ουν μεριμνησητε εις την αυριον). I haven’t read any of the modern literature on this—which I’m sure is very extensive, and very tralatitious—but only Jerome, to whom the word was evidently unknown outside of this particular context. He couldn’t even make up his mind how to interpret it, but the etymology implicit in his hilarious supersubstantialem for the Matthew occurrence surely can’t be right?
See also http://discourse.textkit.com/t/give-us-this-day-our-supersubstantial-bread/13937/6