Gwen Stacy was Peter Parker's first true love.
Next to the death of his Uncle Ben, no death has weighed as heavily upon Spider-Man's shoulders as her passing.
From producer Matt Tolmach:
The relationship between Peter and Gwen is very significant – the previous movies haven’t explored this until now.
Gwen is a very self-assured character; she’s his rival intellectually. And her father happens to be Captain Stacy and let’s be honest, it’s hard enough to meet your girlfriend’s parents for the first time, but when he happens to be the head of the police force that’s chasing you, it makes things that much more complicated.
But there’s an emotional honesty and partnership that’s unique to their relationship. Gwen is really the only person who truly knows Peter – and because of that, there’s a closeness that develops between the two of them that neither of them have with anyone else in their lives.
From director Marc Webb:
One of the reasons why I wanted to use Gwen — first and foremost, she’s his first love in the comics. Let’s just set the record straight, it’s not Mary Jane.
But I like the idea of following somebody who is as smart, if not smarter, than Peter Parker. And Emma Stone is the perfect woman to play somebody who is much more proactive, much more intelligent and feisty.
I just like that dynamic in relationships in movies where they’re kind of lovers as rivals, you know? There’s this back and forth that I love, in the laboratory, and there’s just this great bond that you feel between them. She’s not just a prize, she’s not just a damsel in distress. She’s a confidante, and that was a really important thing.
I'd say using Gwen Stacy instead of Mary Jane also helps distinguish The Amazing Spider-Man from Raimi's Spider-Man.