When subjected to large amplitude oscillatory shear stress, aqueous Laponite suspensions show an abrupt solidification transition after a long delay time t_c. We measure the dependence of t_c on stress amplitude, frequency, and on the age-dependent initial loss modulus. At first sight our observations appear quantitatively consistent with a simple soft-glassy rheology (SGR)-type model, in which barrier crossings by mesoscopic elements are purely strain-induced. For a given strain amplitude γ₀ each element can be classified as fluid or solid according to whether its local yield strain exceeds γ₀. Each cycle, the barrier heights E of yielded elements are reassigned according to a fixed prior distribution ρ(E): this fixes the per-cycle probability R(γ₀) of a fluid elements becoming solid. As the fraction of solid elements builds up, γ₀ falls (at constant stress amplitude), so R(γ₀) increases. This positive feedback accounts for the sudden solidification after a long delay. The model thus appears to directly link macroscopic rheology with mesoscopic barrier height statistics: within its precepts, our data point towards a power law for ρ(E) rather than the exponential form usually assumed in SGR. However, despite this apparent success, closer investigation shows that the assumptions of the model cannot be reconciled with the extremely large strain amplitudes arising in our experiments. The quantitative explanation of delayed solidification in Laponite therefore remains an open theoretical challenge.