To generate random noise in Mathematica based on a given power spectral density Sn(f) or an autocorrelation function, one approach involves creating white noise using Random[NormalDistribution[...]]. After generating the white noise, the next steps include applying a Fast Fourier Transform (FFT) to the noise, filtering it according to the specified spectral density, and then performing an inverse FFT to retrieve the noise that conforms to the desired spectral characteristics. The discussion emphasizes the importance of correctly implementing the filtering process to ensure the output noise matches the intended spectral properties.