The resistance of doped single grain boundaries (GBs) in copper is calculated from first principles and systematically compared to its pure single GB equivalent. As a first step, a state-of-the-art ab initio method is used to calculate the resistivity of doped bulk copper for 16 doping elements at concentration 1 at. %. Results are in qualitatively excellent and quantitatively reasonable agreement with the corresponding experimental data, and allow us to determine Ag, Zn, Mg, Pd, Al, and In as best candidates for GB doping. These atoms have a minimal impact on the bulk resistivity, while they also conform to a set of established criteria for alloying with copper. Then, the specific resistivity of six twin GBs is determined for these elements over a wide spectrum of doping concentrations for the submonolayer and the monolayer GB complexions. Reduced resistivity is observed for Zn, Mg, Al, In, and other elements in two high-Σ GBs, and is qualitatively related to the segregation enthalpy as well as to a low number of empty states around the Fermi energy in the boundary plane region of the GB. The results indicate the possibility for a reduced net resistivity in copper interconnects by GB doping.