Family studies are a useful alternative to twin studies for disentangling genetic and environmental effects on human diseases. However, although age-at-onset traits are often of interest, family-based quantitative genetic analysis of such data is still not commonly used. One reason is that we need multiple random components to capture the genetic and environmental contributions, so it becomes hard to use the existing frailty models for correlated survival data. In this paper we consider the alternative accelerated failure-time models with random effects. The method allows both left truncation and right censoring, and it can deal with an arbitrary family structure and multiple random components. For estimation we use the h-likelihood procedure, which avoids the integration of the random effects in the marginal likelihood approach. To deal with large cohort data, we propose a case-control scheme, where we ascertain all families with at least two events and a subsample of control families. A pseudo-h-likelihood approach is used to analyse the ascertained data. We study the performance of the method using simulated data, and provide an illustration with analysis of melanoma in the Swedish population.