This word may be used either as an adjective, agreeing in gender and case, or as an indeclinable (adverb): LSJ entry at web address 1. The Suda accepts the meaning 'peacefully' (see also alpha 867), which depends on an etymology from a verb of healing (alpha 856) rejected by Aristonicus and Apollonius Dyscolus (schol. to Iliad 4.22) on the grounds that the verb is found only in the middle voice and cannot have an active participle. With the phrase 'from the middle' Suda appears to oppose their argument. In usage, chiefly in Homer, the word seems to mean simply 'wordless(ly), in complete silence', with overtones derived only from the context. Chantraine derives it from ak- 'silence' (alpha 857; cf. V.Pisani, Lexikon des frühgriechischen Epos I 410-1).
The following other etymologies and overtones have been advanced:
(1) 'attentive(ly), giving full attention' (the linguist F. de Saussure, Mémoires de la Société de Linguistique de Paris 7(1892)86 [=Recueil 451], with a plausible etymology from the same root as the Greek verb of hearing, a)kou/w, a)keu/ei; threi= Hesychius, and analogies to Old Norse, Gothic and modern German [aufhören 'to listen and stop'], where the similar words for 'in silence' have lost all sense of 'listen');
(2) 'peacefully, in a healing fashion' (a participle from the verb of healing, so scholia to Iliad 4.22, Hesychius, Eustathius 1902.47, but properly rejected as inapplicable to almost all occurrences of the word);
(3) 'meekly, gently, peaceably' (Boisacq, rejecting de Saussure and citing h)=ka, h)/kistos);
(4) 'in the silence brought on by fear or astonishment' (T scholia to Iliad 9.29a);
(5) 'in the silence of courage brought on by adversity' (related to a)kh/n: alpha 857 n.2).
Of these the first has the greatest merit, for sharpness is a widespread metaphor for hearing and -w- may be another possible extension for ak- (alpha 857), where ak+(extension) -w- behaves differently from the labio-velar consonant -kw-.
[1] From the D scholia to Homer, Iliad 1.512, where the headword occurs.
[2] Homer, Odyssey 21.89.