I can think of three reasons that Shakespeare's works have withstood the test of time. And all of these reasons are in spite of the fact that many of his plot lines were unoriginal; he drew his stories from history, other literature, etc. It's what he DID with the stories that make him brilliant.
First, he creates three dimensional, complicated characters. These characters are very human - with positive and negative attributes - and constantly questioning the world around them. This appeals to any audience (Elizabethan and Modern).
Second, he takes the stories and weaves the plots together in new and captivating ways. He might take ideas from three different sources and put them together in the episodic structure he used; he then weaves a tale that holds interest throughout.
Third, he writes beautiful poetry. The craft of his writing is unprecedented. There are so many quotes of Shakespeare in common use in our society, and some of them, people probably don't even know they're quoting Shakespeare.
All three of these qualities combine to make Shakespeare the great writer we know him as today. I imagine he'll continue to be revered this way for many years in the future.