The present answers are in principle correct, but do not explain the fundamental issues with this idea. In short:
As is mentioned, it is cross-linguistically common for definite articles to develop from deictic markers, which are eventually reanalysed as markers of definiteness. The idea is that the (noun+deictic marker) combination is common; at some point sound changes occur which obscures the distinct building blocks; later the combination is reanalysed as (noun+definite marker+deictic marker); this allows the new definite marker to spread to other configurations. (Of course, the relative positions of these markers can differ.)
This process is much more likely to occur with frequent lexemes. Although "God" is common in the Ancient Hebrew and Arabic texts we have (though surely still not as frequent as deictic markers), this is a feature of the types of texts that were considered worth copying; it is unreasonable to assume that "God" really occurred that frequently in common spoken language.
If that is still not enough evidence, here is the main issue as far as I'm concerned: Arabic and Hebrew split relatively early (after Central Semitic). After Arabic split off, the Hebrew branch (Northwest Semitic) developed several other languages: Ugaritic, Aramaic, and Canaanite (which contains Hebrew, Phoenician, and some other small-corpus languages). Since Aramaic has suffixed -ā, for which it would be rather far-fetched to assume an origin in 'ilāh, and Ugaritic has no definite article, a large part of the development of the Hebrew and Arabic articles must have been independent. This is reasonable when the origin is common cross-linguistically, but an already unlikely development from "God" happening twice makes this even less likely. We would also expect the Aramaic article to be closer to the Hebrew one, since it split off later.
The common derivations of the definite article in Hebrew and Arabic do account for the Aramaic one as well. A good source is Thomas Lambdin, 'The Junctural Origin of the West Semitic Definite Article' (Near Eastern Studies in Honor of William Foxwell Albright, 1971, pp. 315–333). He posits the following stages for Hebrew:
- malku 'a king'; malku ð-(baḥarū) 'the king whom they chose'.
- Close juncture between the noun and postposed words: malku + ð > malkuðð.
- Loss of final short vowels before open juncture leads to malk 'a king', but malkuðð remains.
- h syncope allows this to be analysed as malk 'a king'; malakīm 'kings'; malk hazzi 'this king'; malakīm ha´´illay 'these kings'. In the latter two we recognize ha- prefixed to the deictic markers zi and ´illay.
- This spreads to adjectives, ordinal numbers, etc., and eventually to nouns as well.
In Arabic, the development is similar. Arabic retained final m/n (i.e. malikun instead of malku in stage 1). This final nasal assimilates to the first consonant of the deictic marker in the close juncture. This explains why definite words in Arabic have no final m/n (i.e., why -n is the indefinite article). The definite article becomes analysed as al- because n becomes l before some consonants.
Aramaic differs from Hebrew in stage 4. Lambdin posits that its speakers shortened "this-x and this-y" to "this-x and -y", i.e. malkaððin(āh) wagabraððin(āh) becomes malkā´ wagabraððin(āh). This prevents the reanalysis in Hebrew stage 4 and leads to the postfixed definite article -ā (the "emphatic state").
This leaves Ugaritic to be explained: why no definite article here? The timespan of the Ugaritic corpus is relatively short, maybe simply not long enough to develop a definite article. This may also have stylistic reasons: the definite article is very infrequent in Hebrew poetry which may indicate it was considered not good style for in literature. But also see Na'ama Pat-El, 'The Development of the Semitic Definite Article: A Syntactic Approach' (2009: Journal of Semitic Studies 54:1, 19–50), who argues that Ugaritic displays a premature state of definiteness marking.
