By way of random stab I want to mention a Hesychian gloss:
ἴλαξ· ἡ πρῖνος, ὡς Ῥωμαῖοι καὶ Μακεδόνες
ilax: the holm-oak, according to Romans and Macedonians
(Greek α for Latin e isn't unusual: it was often felt Latin e [ɛ] was closer to α [a] than ε [e].)
If Hesychius (who, it has to be said, lived very late) is right about ἴλαξ being a "native" Macedonian word and not a loan from Latin, this makes it an interesting candidate for the source of the Latin word.
We can't tell if the ι in ἴλαξ is long, but the word has sometimes (though not, I think, by modern etymologists) been connected with Attic ὕλη 'wood', which has a long υ (ὕλη is itself cognate with Latin silva); a PIE preform like *swél-k-s might be constructed. If this is a real word and it has a long initial vowel it could very well enter Latin as īlex (the fact that it looks so much like the many preëxisting Latin words in -ex―even plant names: cārex 'sedge', rumex 'sorrel', ūlex 'heather'―means it slots neatly into the third declension).
Holm oak is a fairly valuable commodity, so presumably the word reached Rome through trade, conceivably through Greek, which wouldn't change it. The earliest attestation of īlex is in Ennius, who was born in Magna Graecia.
The big issue here is obviously that we know very little about Macedonian and its phonological history. Even if Hesychius is right, there are just too many unknowns to be sure.
(If it's related to ὕλη that also explains how it can be an Indo-European word when there are no holm oaks growing on the Pontic-Caspian steppe: the original word just means "(fire)wood".)