When I first started to think about this topic, it seemed to be quite an easy one. There are so many things that could be changed that would improve history, the trick was just to find the worst one, and change it.
Some ideas I had for a re-write of history were: Hitler dying in WWI, Lincoln not being assassinated, no martyrdom of Christ, Trotsky leading Russia rather than Stalin, Columbus not invading the New World, Bush losing to Gore in 2000, South Africa never being colonised, Napoleon winning at Waterloo, the Holocaust never happening (although it wasn’t really a single event), the library of Alexandria not being burnt, or the fall of Rome never happening. 

Most seem like good enough alterations, except for one problem: they are all mutually exclusive revisions.  What I mean is that if one were to be changed (let’s say that Columbus never discovered America) immediately Lincoln could have not been assassinated and Bush could not have beaten Gore (simply because none of those people would have existed).  So by altering history in one way, the future is changed drastically. Even outside of the scope of the single, original event.
Let’s look at another example: Say that Hitler dies in World War One. Everyone goes “yay” right? But can we necessarily say that history will be “better” without Hitler?  Firstly, the feelings of anti-Semitism would still remain in Germany, and secondly by removing Hitler we only open up the floor for a different leader to take over. This leader could either turn Germany into a nation respected and admired for its great policies on human rights, fair treatment and equality for all, or the leader could practice the same genocide as Hitler, possibly with even more ruthlessness and devastation, and instead of losing to the allied forces, go on to beat England and America, succeed in global domination and proceed to wipe out the Jewish race and culture in it’s entirety.  Not good.

So, while I’m sure that no sane person would agree with the actions of Hitler, by simply removing him from history, who knows what we will be opening ourselves up to? 

This is what is known as chaos theory (or more commonly, the butterfly effect).  It says that, if you look at a chain of events in certain systems (for example, the history of the world), the chain is highly sensitive to alterations in it’s initial conditions, and any such alterations would cause change at an exponential rate, causing the system to appear so different in a short time, that it may seem almost random, even though there are no random elements involved. 
In other words, by changing even the slightest aspect of our history, it would be impossible to predict what the world would look like a few years down the line.

It is for that reason that I will rather (for fear of messing up the world) opt not to alter history, but rather to do my best to learn from the mistakes and successes of the past.