Recent lattice Hamiltonian strong-coupling calculations for the (1+1)-dimensional Ising, O(2), O(3), and O(4) models are extended to d+1 dimensions and to general O(n) symmetry. This allows a nontrivial numerical check on the rate of convergence of the strong-coupling expansions as the multidimensional models all have standard second-order phase transitions which have been well studied by other methods. Dlog-Padé analysis of the fourth-order mass gap series for the (2+1)-dimensional Ising model show the critical exponent to be calculated to about the 10% level of accuracy. The behavior of as a function of d and n is exhibited and compared against known results. Mean-field behavior is recovered exactly in the d limit, while the Berlin-Kac spherical model is obtained in the n limit.