According to Perseus Online, ἀκακαλίς is equivalent to νάρκισσος 'narcissus, daffodil'. Some like Jennifer Larson in Greek nymphs: Myth, cult, lore suspect that it could be Pre-Greek but, as always with these sorts of ideas, people are vague about the five Ws. Beekes too proclaims the word Pre-Greek but is vague about what this elusive original word looked like, what it ultimately signified and what precise "Pre-Greek" language we're dealing with. I'm here to suggest something bold and off-the-wall: The word is just plain ol' Greek.
What tipped me off was, for starters, a casual search in the Perseus' Greek dictionary using the search string akak*. I like to begin here when investigating any etymological case labeled Pre-Greek because it makes it easy to find straight-forward native derivations beginning with the same sequence of letters. Fortunately most of Classical Greek's derivations were very regular and the root is normally at the beginning of the word unless a prefix has been attached. If so, one may use the "words containing" rather than "words starting with" tab.
Now in this search list, there was nothing terribly clear to me at first. The word ἄκακος was interesting but it only means 'unknowing of ill, guileless' and it's hard to immediately see why this should be connected with flowers. Yet, a strange instinct compelled me to check this word out further, just in case. Nothing. Then there was another word in the list, ἀκάκης, with two links marked LSJ and Middle Liddell. Clicking first on Middle Liddell, it only mentioned that the word was a poetic form of ἄκακος. Again, zip. So I checked the LSJ entry out of sheer hopelessness. BINGO! There, unassuming and in abbreviated form, it briefly states "epith. of Hades".
That's it! An epithet of Hades, the god of the underworld and death! This harks back to my aptly named post Death and daffodils where I explored a possible native etymology of ἀσφοδελός 'the netherworld asphodel meadow' effectively meaning 'the meadow (ἕλος) not (ἀ-) [reduced to] ashes (σποδός)" or 'unashen meadow'.
So now we have an interesting connection between ἄκακος ~ ἀκάκης 'naive, guileless', Hades and its apparent derivative ἀκακαλίς. Note too that the mythical egoist Narcissus did naively drown to death by his own reflection at water's edge before the narcissus flower (daffodil) rose up in his place, a symbol then of death and rebirth. Thus it's a native word, making Beekes' judgement false in this case.
 Larson, Greek nymphs: Myth, cult, lore (2001), p.187 (see link).
 See Beekes, Greek Etymological Dictionary: ἀκακαλίς.