One great Edward fanfiction is 'Edward's Adventure Beyond the Bebop'. It shows Edward exploring a new planet filled with strange creatures. Another is 'Edward and the Lost Code' which has Edward solving a mysterious code that could save the Bebop crew. And 'Edward's Cybernetic Discovery' is also interesting as it delves into Edward's relationship with her own cybernetic enhancements.
I think 'Edward and the Enigmatic Signal' is a great one. It starts with Edward detecting a strange signal, and as she tries to figure out what it is, a whole mystery unfolds. There are twists and turns as she investigates further. The way it's written really makes you feel like you're right there with Edward.