The evolution of behaviour on islands is a pervasive phenomenon that contributed to Darwin's theory of natural selection. Island populations frequently show increased boldness and exploration compared with their mainland counterparts. Despite the generality of this pattern, the genetic basis of island-associated behaviours remains a mystery. To address this gap in knowledge, we genetically dissected behaviour in 613 F2s generated by crossing inbred mouse strains from Gough Island (where they live without predators or human commensals) and a mainland conspecific. We used open field and light/dark box tests to measure seven behaviours related to boldness and exploration in juveniles and adults. Across all assays, we identified a total of 41 quantitative trait loci (QTL) influencing boldness and exploration. QTL have moderate effects and are often unique to specific behaviours or ages. Function-valued trait mapping revealed changes in estimated effects of QTL during assays, providing a rare dynamic window into the genetics of behaviour often missed by standard approaches. The genomic locations of QTL are distinct from those found in laboratory strains of mice, indicating different genetic paths to the evolution of similar behaviours. We combine our mapping results with extensive phenotypic and genetic information available for laboratory mice to nominate candidate genes for the evolution of behaviour on islands.