We aim to design robotic educational support systems that can promote socially and intellectually meaningful learning experiences for students while they complete school work outside of class. To pursue this goal, we conducted participatory design studies with 10 children (aged 10--12) to explore their design needs for robot-assisted homework. We investigated children's current ways of doing homework, the type of support they receive while doing homework, and co-designed the speech and expressiveness of a homework companion robot. Children and parents attending our design sessions explained that an emotionally expressive social robot as a homework aid can support students' motivation and engagement, as well as their affective state. Children primarily perceived the robot as a dedicated assistant at home, capable of forming meaningful friendships, or a shared classroom learning resource. We present key design recommendations to support students' homework experiences with a learning companion robot.