In this paper, we combine the local linear approach to the relative error regression estimation method to build a new estimator of the regression operator when the response variable is subject to random right censoring. We establish the uniform almost sure consistency with rate over a compact set of the proposed estimator. Numerical studies, firstly on simulated data, then on a real data set concerning the death times of kidney transplant patients, were conducted. These practical studies clearly show the superiority of the new estimator compared to competitive estimators.