This work analyzes normal stress difference responses in blood tested in unidirectional large-amplitude oscillatory shear flow (udLAOS), a novel rheological test, designed for human blood. udLAOS mimics the pulsatile flow in veins and arteries, in the sense that it never reverses, and yet also nearly stops once per heartbeat. As for our continuum fluid model, we choose the Oldroyd 8-constant framework for its rich diversity of popular constitutive equations, including the corotational Jeffreys fluid. This work arrives at exact solutions for normal stress differences from the corotational Jeffreys fluid in udLAOS. More generally, predictions from the Oldroyd 8-constant framework are explored by means of the finite difference method. Finally, the generalized versions of both the Oldroyd 8-constant framework and the corotational Jeffreys fluid are employed to predict the nonlinear normal stress responses the model parameters fitted to udLAOS measurements from three very different donors, all healthy. From our predictions, we are led to expect less variation in normal stress differences in udLAOS from healthy donor to donor, than for the corresponding measured shear stress responses.