I posted this already, but I'm happy to repost. I'm a linguist, and this is what I know: it's not a borrowing, it's a cognate - a native word with cousins in related languages.
In Semitic languages, the original root word for a divine being was *il-, which appears in all branches of the family.
It appears in Akkadian (Babylonian-Assyrian) as il-um, female ilat-um (-um is the same as Arabic al-). There was no god called simply Ilu in Akkadian, though - it was just a word for any god.
It appears in Aramaic-Syriac as Elah-a "God" from *il- + -âh- "great" (-a is the equivalent of Arabic al-
It appears in Hebrew as hâ-El "God" < *il- and Elôhim < *il-âh- + -im. It is only used for the monotheistic divinity.
It appears in ancient Arabian (Northern and Southern) as han-ilahu, feminine han-ilatu; both appear as terms for any divinity (a god/dess) and as proper names (The God; The Goddess). Sometimes you find inscriptions that say "the god of Bosra is Lâh" or "the goddess of al-Hijr is Lât" just like you find ones in Jordan that say "the god of Petra is "He-of-Vegetation (Dhû l-Sharâ)".
In Arabic, it appears as both ilah "god" and as a proper name, Lâh (as in the Shahâda, lâ ilaha illâ al-Lâh.) We also know that Lât - a war goddess - is also mentioned in the Qur'ân along with 3Uzzâ - a protector and mother goddess. Along with "Fate", Manât, they are mentioned by name as false idols.
It is clear from all the various evidences that *Ilat- became *Ilât- because of the parallel with *Ilâh-.
The 7unufâ' and Ahlu l-Kitâb appear to have identified Lâh as "God", which is exactly the cognate word anyway. North Arabian "pagan" religions were very strongly aniconic and monotheistic-oriented, although they paired the main divinity into a married pair. In Nabatæa, the pair was 3Uzzâ' and Dhû l-Sharâ; in other areas, it was Lâh-Lât.