Trapped-ion quantum information processors store information in atomic ions maintained in position in free space via electric fields. Quantum logic is enacted via manipulation of the ions' internal and shared motional quantum states using optical and microwave signals. While trapped ions show great promise for quantum-enhanced computation, sensing, and communication, materials research is needed to design traps that allow for improved performance by means of integration of system components, including optics and electronics for ion-qubit control, while minimizing the near-ubiquitous electric-field noise produced by trap-electrode surfaces. In this review, we consider the materials requirements for such integrated systems, with a focus on problems that hinder current progress toward practical quantum computation. We give suggestions for how materials scientists and trapped-ion technologists can work together to develop materials-based integration and noise-mitigation strategies to enable the next generation of trapped-ion quantum computers.