This paper presents a robust mathematical framework and algorithmic implementation for the design of fractional delay (FD) filters based on the Farrow structure. Although the conventional weighted least squares (WLS) approach provides a computationally efficient design, it typically suffers from prominent localized magnitude error peaks in specific regions, particularly at high frequencies and extreme fractional delays. To resolve this issue, an iteratively reweighted least squares (IRLS) optimization algorithm is proposed. By dynamically updating the weighting matrix according to the error distribution from preceding iterations, the IRLS algorithm steers the $L_2$-norm solution toward a minimax ($L_{\infty }$) equiripple design. Experimental evaluations demonstrate that the proposed method suppresses the peak magnitude error by 8.1 dB, yielding an approximately equiripple error surface across the frequency-delay plane.