For this headword -- actually the present participle, feminine nominative singular, of αἴθω (alphaiota 140) -- see generally LSJ s.v.; it thus conveys the idea of a part of the (Homeric) house designed to catch the sun and so be illuminated.
Adler cites the scholia to Homer, Iliad 6.243, where αἰθούσῃσι occurs (see web address 1), and Hesychius s.v. αἰθούσῃ , quoted from Iliad 9.472.