Friction stir spot welding (FSSW) is one of the most popular fusion joining processes. The process is a solid-state welding process that allows welding of weldable as well as non-weldable materials. As a part of this investigation, weld samples of Al6061-T6 were reinforced with silicon carbide (SiC) powder with an average particle size of 45 µm. Initially, a Taguchi L9 orthogonal array was developed with three factors, i.e., rotational speed of the tool, pre-dwelling time, and diameter of the hole that was filled with SiC before welding. The effects of the SiC particles and process parameters were investigated as tensile–shear load and micro-hardness. The optimisation of parameters in order to maximise the output responses—i.e., strength and hardness of the welded joints—was performed using a hybrid WASPAS–Taguchi method. The optimised process parameters obtained were a 3.5 mm guiding hole diameter, 1700 rpm tool rotation speed, and 14 s of pre-dwelling time.