This is the first study reporting a full three-dimensional (3D) non-normal-mode onset of convection in a porous medium. In this paper, the onset of thermal convection in a vertical porous cylinder is investigated theoretically. In particular, the contribution includes the following novelties. The homogeneous cylinder has a circular cross section. The eigenvalue problem is non-separable in space because of a constant heat flux condition at the upper boundary. In addition, a partly conducting cylinder wall is represented by a Robin parameter a. All boundaries are impermeable. The eigenvalue problem is solved numerically in the COMSOL Multiphysics environment. The critical Rayleigh number for the lowest onset modes is reported as a function of the ratio of the cylinder radius and its height. The only mode number, m, represents azimuthal dependency. The axisymmetric mode m=0 corresponds to the preferred mode of convection for a small-cylinder radii. The numerical onset criterion is validated with the well-known analytical limit case, a ->infinity. Finally, we performed a visual comparison of the thermo-mechanical eigenfunctions against an established problem, where the only difference is the thermal condition at the upper boundary. The present 3D analysis is a step toward a full adequate modeling of experimental reality for convection onset, beyond the standard constraints of mathematical convenience.