All He had was a bow and arrow
Kate Bishop could have been any hero. When she joined the Young Avengers she didn't have a family legacy to hold up or powers that were just like another Avenger's powers. She had been learning combat fighting and self-defense in several forms including archery, swordplay, and martial arts so when she raided the Avengers mansion there was no reason she couldn't take after Mockingbird or even Black Widow, super heroines whose powers are subliminal. So why would she choose to emulate Hawkeye?
I for one think there are a few reasons. The first is that Clint is human. Clint has always been open about being a regular guy and in a world where most heroes are known for their super powers. Kate was a normal girl who was trying to enter a world of young heroes whose powers are beyond anything she could think of. Like their adult counterparts they needed someone who could put things in perspective the way Clint Barton had always done for The Avengers.
The second, and less serious reason is the color purple. Kate, even before joining the Young Avengers was constantly wearing Hawkeye's signature color. Perhaps she was drawn to the bow and arrows because they reminded her of a hero who wore the same color she did. Perhaps though, it's more important to consider what might have started Kate's love for the color.
Finally, I think that the most significant and obvious reason why Kate Bishop chose to become Hawkeye is because he was her role model. When Kate was a child she was kidnapped by the villain El Matador when she was a child because she had witnessed her father making dirty deals with the villain. Kate managed to escape Matador but was almost recaptured when Hawkeye and the other Avengers arrived to save her. Hawkeye was the one who kept Matador from catching her though and Hawkeye was the one who had been watching El Matador and knew to rescue Kate.