Using density functional theory, we study physical properties of boron nitride nanotubes (BNNTs) with the substitutional carbon pair defect. We also consider the Stone-Wales (SW) rearrangement of the C-C pair defect in the BNNT. The formation energy of an SW defect of the carbon dimer is approximately 3.1 eV lower than that of the SW-transformed B-N pair in the undoped BNNT. The activation energies show that the SW defect in the C-doped BNNT may be experimentally observed with a higher probability than in the undoped BNNT. Finally, we discuss the localized states originating from the carbon pair impurities.