This paper considers the thermo-convective boundary-layer flow (BLF) of a water–copper mono-nanofluid over a flat vertical surface which is subjected to three types of periodic temperature variations described by the sinusoidal, sawtooth, and triangular waveforms. The temperature of the fluid at the flat surface is greater than the surrounding ambient temperature. The governing equations describing the BLF have been reduced to a non-similar form using an appropriate stream function formulation. The Keller-Box method is used to obtain numerical solution of the boundary-value problem. The effect of the pertinent parameters on the nature of the flow and the heat transfer has been discussed using actual thermophysical data. The results about the shear–stress and heat transfer rate at the surface are presented as well. To study the nature of BLF, the velocity and thermal boundary-layers, the streamline and isotherm plots have been considered, which reveal that the nanoparticle volume-fraction, amplitude of surface temperature variations, and the Grashof number play a pivotal role in enhancing/diminishing heat transfer. The final outcome reveals that the heat transfer is highest for the sinusoidal waveform, followed by that of the triangular and then, the sawtooth. An important inference is that a symmetric periodic temperature distribution at the surface enhances heat transfer more than that of a constant surface-temperature.