Materials derived from copper oxide, Cu2O, are potential p-type transparent conducting oxides. Cu2O displays some unusual features, including p-type semiconductivity and the importance of cation-cation interactions in determining the band gap and transparency. We apply first-principles density functional theory to investigate how dopants with a range of ionic radii, oxidation states, and electronic structure can be used to tune the band gap. Unlike many oxides, in which the band gap is reduced by the appearance of dopant induced states in the host band gap, the band gap Of Cu2O can be both increased or decreased by a suitable choice of dopant. Two effects dominate: (i) dopant-induced changes to the Cu-Cu interactions through structural distortions around the dopant site and (ii) the alignment of the dopant electronic states with the valence band or conduction band of Cu2O. Dopants with ionic radii larger than Cu+ (Ba2+ Sn2+, Cd2+, In3+, La3+, and Ce4+) produce strong structural distortions around the dopant site. Dopants with ionic radii smaller than Cu+, such as Al3+, Ga3+, Ti4+, and Cr4+, show no structural distortions. Structural distortions disrupt Cu-Cu interactions and for Sn2+ and La3+, this opens up the band gap, potentially improving the transparency. However, if dopant electronic states interact with the valence or conduction band of Cu2O, e.g., In3+ or Cd2+, or produce defect states in the Cu2O band gap, e.g., Ce4+, the band gap is reduced, regardless of dopant-induced disruption of the Cu-Cu interactions. We present a set of materials design guidelines to be used for choosing potential dopants in Cu2O.