Felicia Hardy/Black Cat
Felicia Hardy is one of Peter Parker's more famous love interests, and though she has shown up on film before (played by Felicity Jones in The Amazing Spider-Man 2), she wasn't given very much screentime or a proper adaptation - which makes her a great candidate for this new franchise.
Hardy is the daughter of a renowned cat burglar and is sexually assaulted during her freshman year of college. Refusing to become a victim, she becomes the Black Cat, and follows in her father's footsteps.
Felicia eventually falls for Spider-Man and even strikes up a romantic relationship with Peter Parker (neither one of them knowing the other's identity, of course).
While introducing the Black Cat this early in the franchise may be hard to do, bringing Felicia in early on would be wise and could greatly complicate Peter's love life, again adding to the soap opera elements that Feige and co. are aspiring to.