There's a saying I've heard writers throw around, "There are no new plots, just new characters." Some writers even argue that there were never that many plots around to begin with (see Christopher Booker's The Seven Basic Plots).
I agree with both sentiments, and I think the writer's struggle for an "original" story is both futile and unnecessary. In most cases, the characters are what make a narrative feel new and interesting, not the plot.
However, this brings up another issue: there aren't really any new characters, either. Just like Booker's basic plots (Quest, Comedy, Tragedy, Journey and Return, ect.) there are also basic character archetypes (hero, villain, wise-man, mother figure, ect), and they've all been run through the fiction grinder time and time again.
How do you make an old archetype seem fresh? Details. Ever since The Odyssey, people have been telling stories where the hero has a tragic flaw (Macbeth, Jay Gatsby, Don Draper). These characters are still compelling because they are richly drawn— they're given their own quirks, tics, and unique experiences. Without these details, the worn-out and tired nature of this archetype crawls into view.
I feel like this is a subjective topic, though. Different people connect with different parts of stories. For example, I thought Avatar was flat-out boring because the characters were about as unoriginal as humanly possible, and I couldn't get past that. Yet, loads of people didn't care because the world James Cameron created was compelling to them.
So, which part of a story do you connect with the most? Plot, setting, character, theme, premise? Some other part I didn't mention. It's temping to say a mix of everything, but see if you can nail down a #1. Mine's character, all the way.