I'm fairly new to AH.com, so I don't know if there's a dedicated thread for ideas of alternate timelines, nor do I know if the following ideas have already been proposed or published. Please let me know if either type of thread (general ideas, or one of these specific timelines) exist.
[Also, I first posted this to the AH Writer's Forum, but realized this forum would probably be a better fit, so I apologize].
As for the ideas themselves:
1) Ancient Israel Invicta.
The two great Kings of Israel, David and Solomon, unified the land of Palestine against Philistine incursions and made Israel into one of the most influential kingdoms in the region. What if the Kingdom was not split after Solomon's death, and continued to increase in power?
Given that the Bible records Solomon as having strong ties to the Phoenicians, allying with them to provide the resources for building the first Temple, it wouldn't be hard to envision that alliance strengthening until it resembles a 'Union' a la Denmark-Norway or Poland-Lithuania. Then Israel would have a base from which to colonize or influence much of the rest of the Mediterranean, including Greece, possibly including Carthage, which means it might later butt heads with Rome.
[Edit: I wonder if King Solomon might be inclined to build up Israeli industry to some degree, which would mean 'King Solomon's Mines' might not be so apocryphal in this TL.]
Short term, Israel would have to face off against incursions from Egypt (a successful campaign there might give Israel control over the Nile delta or even the lower Nile region) and Assyria (among other Mesopotamian empires).
With sufficient expansion, you might see an earlier development of the Hebrew faith from a Temple-centric religion to something approaching modern Judaism, centered around local synagogues. It'd also be interesting to see how the expanding borders and power of a Israel-based empire intersects with the Jewish tradition of a Messiah from the line of David, who will bring forth the kingdom of God. Due to the Babylonian conquest and Exile of the Jewish people, that Messianic promise was largely assumed to refer to temporal (physical, political) power, which meant that Christianity re-interpretation of the Messianic texts as speaking of a spiritual kingdom of God came as quite a surprise. Perhaps this ATL interpretation would more closely resemble the Christian/Catholic idea -- if Israel already has political power, an unrealized future 'kingdom of God/heaven' would have to be something new, something unseen.
2) Earlier Origin of Classical Economics
I've seen a few threads that hypothesize the survival of one of the great commercial empires of the medieval world (such as the Republic of Venice, or the Hanseatic League). While those are admittedly interesting, I'd like to propose a somewhat broader point of divergence.
If you asked them when 'economics' began as a subject, most people would point to the Scottish philosopher Adam Smith. The reality is, Smith really didn't contribute that much -- most of his observations were taken from the French Physiocrats (who are severely misrepresented by most history texts, if they are even mentioned at all), who owed much of their ideas to earlier theorists arising out of the High Middle Ages and the Renaissance.
The problem was that, for much of medieval and modern history, economic policy for virtually every nation was essentially mercantilist in nature --they centered around increasing the coffers of the state (bullionism), increasing land for agricultural use (agrarianism), increasing population (labor theory of value (and increasing national self-sufficiency (favorable balance of trade).
It wasn't until Adam Smith came around that these ideas started taking root in England, which was one of the earliest nations to even consider rejecting mercantilism. It wasn't until even after that, with Jean-Baptiste Say and David Ricardo, that classical economics developed into something we would recognize today, with Say's Law of Markets, Ricardo's theory of comparative advantage, such concepts as 'entrepreneurship,' etc.
Now, much of this theoretical advancement may be contingent of historical events. The High Middle Ages bore witness to many developing institutions, such as banking in Italy, bond markets in Venice, stock exchanges in Amsterdam, etc. Many times it was personal experience with such institutions that inspired various economists to come up with their theories. But not always, and that's what I'd like to explore.
What if... there was an earlier development of the seminal notions of modern economics? Specifically I'm thinking of two elements: first, the acceptance of industry and commerce as essentially productive elements in society (contra the nearly universal notion that only agriculture is productive), and second, the concept of comparative advantage (Ricardo's theory that if one nation specializes in good A, and another nation specializes in good B, and they trade, they will both be better off than they were before -- i.e., have more of both good A and good B than when they started).
If these theories were discovered earlier (perhaps pushing forward others as well), what effect might that have? Well, the French philosopher Montesquieu argued that nations that trade together extensively tend to trust each other and profit from each other and therefore don't go to war with each other as often (lest they risk that trust and profit). Erasing the social taboo against non-agricultural work would allow for a more rapid growth of the middle class (diverting resources from warfare to commerce and industry, and ensuring that bored nobles have an outlet for their restlessness besides constant warfare).
And, as mentioned above, sophisticated economic theories might enable various commercial republics (e.g., Venice) or leagues (e.g., the Hanseatic one) to optimize their domestic and foreign policies, enabling them to flourish even in the face of increasingly powerful nation-states after the treaty of Westphalia. In other words, who knows, but it'll be a wild ride.
3) ATL Voyage of Saint Brendan
I've read a number of books that indicate that Columbus' single-minded pursuit of lands across the Atlantic may have been inspired by his reading of 'The Voyage of Saint Brendan,' a rather legendary Irish monk who was born c. 500 AD. The 'Voyage' tells of Brendan's adventures at sea with a dozen or so other monks, in which they encounter many lands and fantastical creatures. Some modern readers have argued that the 'Voyage' actually describes a round-trip from Europe to America and back.
Let's concede for the moment that this interpretation is correct, that the Irish monk did discover America, and later inspired Columbus. Here's the divergence point: what if... Brendan never returned, and his 'Voyage' was never written? Brendan was known for having previously circumnavigated Ireland, founding new monasteries and Christian communities wherever he made landfall. What if he took the same approach when he discovered America?
Even if his crew was limited solely to himself and a dozen other monks, Brendan might still have an influence on later history. Imagine an Irish monastery planted on the shores of the Potomac, Brendan and the others hard at work learning the native languages to convert the locals. Imagine him teaching literacy to the locals, transcribing by hand a copy of the Bible... and then perhaps a few works by Plato, Cicero, or other Greek or Latin authors. Imagine if he tried to recall and teach various technologies that he'd witnessed from Europe. And imagine if one of the monks in Brendan's crew carried a dormant form of smallpox or some other horrible disease....
All of these elements would be incredibly disruptive to native American society, but it would given them several hundred years to regroup before European began to colonize the world (perhaps longer if, as we posit, Columbus was no longer inspired by Brendan...)
Another thing to consider is that none of these changes require Brendan or the others to be able to marry and reproduce. I don't know canon law as it applies to monks in sixth century Ireland, but I'm pretty sure all monks took a vow of chastity. But I don't know if that would have applied to priests at the time, and I know it would have applied to lay Christians who decide to join a pilgrimage. What if Brendan had sailed a bigger boat, or sailed several all together, and other Christians came with him? Instead of just founding a monastery, they might found a small town. Would they avoid the fate of Roanoke, and create a quasi-European colony on the shores of North America, five hundred years before the Vikings and a thousand before Europeans expand westward in force?