Hugo's obsessed with it. His father tried to fix it, and Georges Méliès made it. From the beginning of the book right until the end, this object is the center of these folks' universe because it connects them to a happier past:
Increasingly, Hugo felt like he had to try. If he fixed it, at least he wouldn’t be so completely alone. (1.5.54)
When Hugo finds the automaton, after his father’s death, it reminds him of their old life together, with his dad tinkering with things and telling his kiddo about the movies.
In a way, Georges Méliès has a similar relationship to the automaton, just not at first:
“I was haunted by those ghosts for so many years. The only thing I couldn’t bring myself to destroy was the automaton.” (2.8.17)
Méliès is pretty ashamed of his past. The automaton is the one thread to Méliès’s history that he lets survive, and when Hugo finds it, it brings the past back to him—in a good way. He remembers that the past wasn't all bad—there were times when he was a great inventor, a creator of magic, and an innovator in the truest sense.
It's a good thing Méliès didn't destroy the automaton, because it ends up being just the thing to bring this new family together. It all starts with Hugo's papa, who finds it at the museum and decides to fix it. Then Hugo takes up the torch when his father dies in the fire. And when he turns Isabelle's key in the back of the automaton, that's what launches them on the path to discovering the truth about Papa Georges.
And when Georges discovers that Hugo has had it this whole time, well, that's what launches them on the path toward becoming a family, because Georges can finally come to terms with his past and move forward. It almost seems like every key event in the novel is set into motion because of this tiny toy.
Plus, the automaton has a magic all its own. Way back in the day, automatons were pretty incredible technology. The fact that Georges Méliès could make this little toy write was quite the feat for his day and age. But then, when Hugo and Isabelle fire the thing up for the first time, it doesn't write, it draws. And it doesn't just draw doodles; it draws a detailed scene from George's movie. Pretty impressive, no? So even for us, the automaton is a symbol of the power of imagination (Georges's) to create magic (the automaton's automation).