Ok, the idea is the following:
If condition A makes B possible and B makes C possible then A => C
Example:
If Sergey Koroliev had survived his cancer, the Russian moon landing project would have worked, in consequence the American moon missions would have continued (and we would have moon bases now).
Try to make them plausible (like conspiracy theories ;)).
If all females were sterilised during a decade, there would be a maxiimum ten yearsworth gap in babies giving a potential maximum of ten yearsworth savings on state pensions and more money to spend on the rest of the elderly.
A British squadron was unable to catch battlecruiser Goeben before reaching Turkey due to mysterious communication problems (with the authorities in London). Had Goeben been stopped, Turkey would have stayed out of WW1 or joined the Entente. The Entente Powers could have supplied Russia and would have had no need to divert forces from the Western front. WW1 would have ended 1-2 years earlier, i.e. before the Russian revolution, which then would not have taken place.
=> a lowly clerk in a London office caused the Soviet Union.
If the invincible army had attacked under better weather Elizabeth I would have been overthrown and the age of pirates (and possibly the colony of Virginia) could have been averted (plus all the consequences of a Catholic England). ;)
If the English Army under King Harold had not force marched north to defeat the last serious Viking Invasion of England at Stamford Bridge, forcing them to make another forced march of 250 miles immediately after the battle to meet the invading army of Duke William of Normandy at Hastings, then they would easily have defeated Duke William's army. Then the whole history of Western Europe would have been substantially different. No Robin Hood, no Plantagenets, no New Forest, no Tudors, no Stuarts, no English Language as we know it, no Shakespear...
Following Mero's lead:
Had the wind not changed direction for another week (or even less), William would have run out of supplies and would have had to abort his expedition. Consequences, see above (plus no Bayeux tapestry).
If Lady Houston (http://en.wikipedia.org/wiki/Lucy,_Lady_Houston) had spent her money on gardening instead of aero engines then we'd have been a few important designs short and Adolf Hitler would have worn a kiss-me-quick hat (http://www.telegraph.co.uk/news/uknews/4782054/Adolf-Hitler-saved-Blackpool-from-Blitz-because-he-wanted-it-as-a-personal-playground.html).
If Hitler had died in a car accident in 1938 (he nearly did), then he would still be considered a great statesman with just a slight character flaw (his rabid antisemitism). The likely downfall under his successors would not be attributed to him.
If Lincoln had decided not to declare war on the south and let the secession happen, the US would likely not entered WWI, letting Germany and Austria-Hungary win, hence avoiding WWII.
If Ogedai Khan]http://en.wikipedia.org/wiki/Ogedai]Ogedai Khan (http://en.wikipedia.org/wiki/Ogedai) had not been an alcoholic then he would have lived much longer and the invasion of Europe would not have stopped after the Battle of Liegnitz. Since the impact of Mongol invasions was huge, European history would have taken a completely different course.
If Christianity had not showed up, Rome's decline would have been ameliorated and the middle ages would have been averted probably having a industrial revolution happening 500-800 years earlier.
I doubt that. The invading barbarians were also partially Christian and they invaded not because of that.
To the contrary, the empire became unstable on its own accord and the breaking apart was slowed for some time by the common faith.
The fatal effects of Christian dogmatism began to show later. I think the church was responsible for cementing the lower status quo after the downfall without being directly responsible for it.
My opposing proposal:
If the electrical telegraph had been invented at the time* of the good emperors (i.e. before Commodus) then it would have been possible to keep a centralized administration of the empire. The existing slow communications required de facto independence of the different parts leading in the long run to actual independence.
*imo technically possible
But when you see what happened to Alexandria the effects of Christianity became obvious. Certainly climatic changes pushed the barbarians, and on the same token Christianity had no relation with the Huns, but the intellectual health of the empire was definitively affected by Christianity. I don't know if the empire itself would have collapsed or not (the eastern part survived one thousand years) but it is likely that the technical knowledge would have been preserved in Alexandria, Greece, and Constantinople even if the barbarians teared the west apart.
Now, it is perfectly possible that the industrial revolution would've been postponed anyway given that slavery was prevalent and cheap, and you could argue that Christianity had a role in that change of attitude, but it wasn't until the Illustration (which wasn't particularly religious) that the pressure for a technical alternative of production became necessary.
I think the most important part (apart from the 'too big to be governed') was the unwillingness of the rich to take part in the common defense (either as soldiers or by paying taxes for it). As a result mercenary barbarians were hired and paid to fight other barbarians until they found out that it was more lucrative to take over themselves (and to hire yet other barbarians to defend the borders...wash, rinse, repeat until at least the Western part was history). Barbarians were cheaper and the better fighters. The original Roman superiority was not in the individual soldier's skill (tribal warriors were far better) but in the organization and discipline.
There are ominous signs that the US follows a similar path.