Einsiedler, Mozes, Shah and Shapira [Compos. Math. 152 (2016), 667-692] prove an equidistribution theorem for rational points on expanding horospheres in the space of d-dimensional Euclidean lattices, with d 3. Their proof exploits measure classication results, but provides no insight into the rate of convergence. We pursue here an alternative approach, based on harmonic analysis and Weil's bound for Kloosterman sums, which in dimension d = 3 yields an effective estimate on the rate of convergence.