This is a great question and several great, and quite accurate, responses have been supplied which outline specific instances and their traces of growth.
Since those have been supplied, I will address the more loose range of the subject which targets the growth of civilization and how such a motion occurs to begin with.
As some have already alluded to in some fashion or another, the move to some resembling form of monotheism typically occurred for political reasons.
As it has been noted, effectively all civilizations were theocratic in the Bronze Age and was still quite common in the Iron Age.
When we look at the evolution of human civilization around the Mediterranean region, what most notably stands out is the evolution from clusters of people to City States to unified City States to Kingdoms; not always in perfect line and with some exceptions to this construct (for example, Egypt is hard to determine in this manner as it was well formed into a full Kingdom by a point far earlier than surrounding cultures which we can far more easily observe in regards to early evolution).
In every case, however, the theocratic motive (Roman adoption is similar but different) for the monotheistic solution is a change from the ruling power being that of an individual pleading to the deities in some fashion, typically through a political party of priests, into a single source system where the ruling power appears to do less pleading and more "channeling" (we'll borrow that phrase to summarize) of some selected/reformed deity (somewhat like we think of the Pope today, but not exactly the same as each culture is its own form).
Interestingly, these movements don't seem to last with near pure unilateral power intact, but the religious summation doesn't appear to redistribute post-unilateral ruling power once power has been redistributed.
Loosely speaking, and with much error in specific as this is a generalization, some party charges the ruling unilateral power of some wrong and the system of government is diversely shifted in some manner (usually to a two-branch system which typically previously already existed) and the communal branch (priests, or political senate style) takes some of the powers related to the theocratic associations typically revolved around law making and the "King" (in the various forms) takes (what we might call today) "executive" controls and powers.
I am unaware of any event whereby such occurs and the subsequent reaction following the charge of corruption results in a dissolving of the monotheistic-like religious alterations.
Rome is a bit different due to its later and more evolved political infrastructure, but it more or less falls along these lines to a lesser degree.
It is easy to imagine the motive is greed, and in some manners we may say such is the case, but with that greed we should also note that in each case a very real political and social issue was being addressed; each unique to their culture and time.
However, as a very generalized view, we can understand that in each early case the ruling power was seeking to unify some perceived schism, end political conflict and in-fighting, and gain capabilities which would allow them to restructure their ruling area's infrastructure into one singular design and focus.
In a way, we could say these were "experiments" for solutions to issues that arrived as populations grew and the dynamics of party interests became such that a range of diverging ideas and visions about how to proceed "nationally" were perceived to be a problem; a weakening.
Effectively, one could say the idea of "categorization" was the basis for the change, as cultures continued to struggle to survive; the solution for survival was tried in multitudes of manners of diversities, with a constant focus reappearing for a singular cultural integrity - a unification and singular front.
It is quite worth noting that in most cases (all that I am aware of, but I may be missing an exception) the "monotheistic" movement politically arrives as part of a solution during extreme civil stress within a theocratic society engaged in polytheistic diversities.
This is itself not a foreign behavior for humans, to unify power and direction during extreme stress.
Take, for instance, Dwight D. Eisenhower and the surrounding political commotion regarding the notion for a Supreme Commander to address the issues presented to the Allied forces in retaining diversified command.
This event of new and stressful circumstances which appear to be beyond the ability of any to solve with the "current" paradigm and therefor permitting the argument and conclusion of a unilateral command shares a sociological relation to the shifting of theocratic civilizations from diverse pantheons with diversified governing to a unified deity and political ruling.
Like Eisenhower, too, again, each of such motions tends to be short-lived before the unilateral political power is once again diversified; though the ideologies outlining a unified culture during that time appear not to be thrown out (again, similarly, the Allies stopped being "governed" by a unilateral commander, but the idea of such command and unification in military global control greatly influenced the operational procedures existent today in the UN).
That's my two cents
Cheers,
Jayson