This paper considers the problem of robust three-dimensional (3-D) angle-of-arrival (AOA) source localization in the presence of impulsive α-stable noise based on the I_p-norm minimization criterion. The iteratively reweighted least-squares algorithm (IRLS) is a well-known technique for solving I_p-norm minimization with the desirable global convergence property. Adopting the IRLS for 3-D AOA localization requires nonlinear-to-pseudolinear transformation of azimuth and elevation angle measurement equations, thus resulting in a new variant of the IRLS, called the iteratively reweighted pseudolinear least-squares estimator (IRPLE). Unfortunately, there exists correlation between the measurement matrix and noise vector in the pseudolinear measurement equations, which consequently makes the IRPLE biased. To counter the bias problem of the IRPLE, a new iteratively reweighted instrumental-variable estimator (IRIVE) is proposed based on the exploitation of instrumental variables. The IRIVE is analytically shown to achieve the theoretical covariance of the general least I_p-norm estimation. Extensive simulation studies are presented to demonstrate the performance advantages of the IRIVE over the IRPLE as well as other existing least-squares and least I_p-norm estimators. The IRIVE is observed to produce nearly unbiased estimates with mean squared error performance very close to the Cramér-Rao lower bound.