Organ donation has a crucial impact on patient care and survival, of which the worldwide gap between organ demand and supply is currently one of the most challenging issues. Brain-dead patients are the main source of organs that can be donated, but donation requires the consent of family members—a choice that is often complex and stressful and leads to refusal of consent. This mini-review aims to provide an overview of the current knowledge on the impact of certain psychosocial factors on the decision-making process with regard to organ donation by family members. In particular, the influence of several aspects is emphasized, such as sociodemographic factors, knowledge of the organ donation process, religious beliefs, concerns that are related to the choice to donate, and mode of communication. Consistent with this evidence, we emphasize the need to examine these aspects further through interventions and guidelines that improve the organ donation application process and ensure a positive experience for the family that has to make the decision.