Besides Eddie, who is the next most important character in Buried Onions, and what's so important about him or her?

I would say another important character is Eddie's coach.  The coach believes in him, despite the problems that exist in the barrio (drugs, crime, etc.).  The coach acts as a mentor, friend, brother, and father figure.  Eddie says, "He'd known all of us glue sniffers—Angel, Lupe, dead Jesus, and me—since we were on bikes with training wheels. He'd known that most of us would mess up, some more than others. Still, it pained him."

The coach provides hope and dignity when others outside the community assume the boys are both doomed and useless.