Hmmm it is yours so its up to you, but I'll still offer a reccomendation...
Why not just ask a couple friends? Give your friends an overview of the story, what you plan to do with it, and brainstorm a bit. Ya'll might come out with something intresting and unique, or you could just name it something crazy, like The anime Bleach for example. Hope that helps
Keep it as blackbird, don't worry about it being too common. I personally think the fact that it's not revolving around the character Blackbird much makes it a better title. It bothers me when something is named after the main character or something, or like when a song's name is the chorus of the song. I like it better when it's something that's only mentioned even once or something, it somehow makes it more special that it's named that.. in my mind, at least. I am a little strange though, so you never know.
Thanks especially for giving the explanation on the story. But I voted for keeping it Blackbird. People use "Raven" too much on their stories and yours seem more intriging. So especially if it's a common theme of the story, you should keep it as is. *nods*
Since I haven't read the story, I can't make an executive decision on the name. What I know is you should keep it simple but not too cliché, because that's always the best way to go. Blackbird is nice too, though ^_^