Strong interaction with other particles or feedback from the medium on a Brownian particle entail memory effects in the effective dynamics. That motivates the extension of the fluctuation-dissipation theorem to nonequilibrium Langevin systems with memory. An important application is to the nonequilibrium modification of the Sutherland-Einstein relation between diffusion and mobility in the case of strong memory. Nonequilibrium corrections include the time correlation between the dynamical activity and the velocity of the particle, which in turn leads to information about the correlations between the driving force and the particle's displacement.