Heart, Conformation and Training.
I personally don't base too much off the bloodlines. The 1D barrel horse that I'm leasing for 2012 cost her owners $50 for the stud fee. She's grade and 14 hands. She's had great, slow, consistent training...she's got a huge heart and a love for running and giving her all, and she's built like a brick house. She doesn't have the greatest conformation, she's extremely downhill, but she's all muscle and is powerful.
My mare, has decent bloodlines...She does have some racehorse breeding to her, quite a few AA and AAA horses in there. And she is a money winner too. Her times are just off of the grade mare I ride, she could probably beat her if she's conditioned right.
I'd prefer to have a horse with a heart of gold that has the will to win, the conformation to get us there...I can't have a unsound horse, and the correct training on it. Bad training is very noticeable, we've all seen it, and I don't want to be "that person". You can have the fastest horse in the world, but until it's trained right, you won't get anywhere on it.
Having a horse that is bred to run ontop of the other 3, is a bonus in my book.