The best evidence that Abraham Lincoln subscribed to the idea of American exceptionalism can be found in the Gettysburg Address.
American exceptionalism is, basically speaking, the idea that America is different from all other countries in some important way. Lincoln shows this belief in his speech. He does so by portraying the US as a noble experiment, one which will prove whether freedom and democracy are possible.
He says that this was a nation that was
conceived in liberty and dedicated to the proposition that all men are created equal.
By saying this, he is implying that the US was different from all other countries that were not dedicated to such ideas.
In addition, Lincoln is saying that the US is really the only hope for the world. He is saying that the kind of government and society that the US has is something that is different from (and better than) what everyone else has. For this reason, it is up to Americans to make sure that
government of the people, by the people, for the people shall not perish from the earth.
By saying such things, Lincoln is saying that the US is exceptional in that it is the only real hope for humankind.