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Recent experiments show that quasi-one-dimensional lattices of self-propelled droplets exhibit collective instabilities in the form of out-of-phase oscillations and solitary-like waves. This hydrodynamic lattice is driven by the external forcing of a vertically vibrating fluid bath, which invokes a field of subcritical Faraday waves on the bath surface, mediating the spatio-temporal droplet coupling.By modelling the droplet lattice as a memory-endowed system with spatially nonlocal coupling, we herein rationalise the form and onset of instability in this new class of dynamical oscillator. We identify the memory-driven instability of the lattice as a function of the number of droplets, and determine equispaced lattice configurations precluded by geometrical constraints. Each memory-driven instability is then classified as either a super- or sub-critical Hopf bifurcation \emph{via} a systematic weakly nonlinear analysis, rationalising experimental observations. We further discover a previously unreported symmetry-breaking instability, manifest as an oscillatory-rotary motion of the lattice. Numerical simulations support our findings and prompt further investigations of this nonlinear dynamical system.