Adler cites the unedited Ambrosian Lexicon (712) as a source for the entry, and it is taken over by ps.-Zonaras.
The headword verb, otherwise unattested (and not in LSJ), is, as Chantraine suggests, evidently derived from ai)/linon ('mournful cry', 'dirge': alphaiota 180, alphaiota 181, alphaiota 182); ai)liniw= and e)liniw= would be homophonous in the Byzantine period. A different verbal derivative from this noun has been conjectured in Dosiadas' poem Altar (Greek Anthology 15.26.15): ai)lineu=nt[a], where most mss have e)ll-.
Though the headword has been translated here as a present verb since that seems to be how the lexicographer understood it, it is perhaps more probably future tense from a hypothetical *ai)lini/zw.