The recent development of high-throughput sequencing technologies calls for powerful statistical tests to detect rare genetic variants associated with complex human traits. Sampling related individuals in sequencing studies offers advantages over sampling unrelated individuals only, including improved protection against sequencing error, the ability to use imputation to make more efficient use of sequence data, and the possibility of power boost due to more observed copies of extremely rare alleles among relatives. With related individuals, familial correlation needs to be accounted for to ensure correct control over type I error and to improve power. Recognizing the limitations of existing rare-variant association tests for family data, we propose MONSTER (Minimum P-value Optimized Nuisance parameter Score Test Extended to Relatives), a robust rare-variant association test, which generalizes the SKAT-O method for independent samples. MONSTER uses a mixed effects model that accounts for covariates and additive polygenic effects. To obtain a powerful test, MONSTER adaptively adjusts to the unknown configuration of effects of rare-variant sites. MONSTER also offers an analytical way of assessing P-values, which is desirable because permutation is not straightforward to conduct in related samples. In simulation studies, we demonstrate that MONSTER effectively accounts for family structure, is computationally efficient and compares very favorably, in terms of power, to previously proposed tests that allow related individuals. We apply MONSTER to an analysis of high-density lipoprotein cholesterol in the Framingham Heart Study, where we are able to replicate association with three genes.