We report the first measurement of the target-normal single-spin asymmetry in deep-inelastic scattering from the inclusive reaction 3)He(↑)(e,e')X on a polarized (3)He gas target. Assuming time-reversal invariance, this asymmetry is strictly zero in the Born approximation but can be nonzero if two-photon-exchange contributions are included. The experiment, conducted at Jefferson Lab using a 5.89 GeV electron beam, covers a range of 1.7<W<2.9 GeV, 1.0<Q(2)<4.0 GeV(2) and 0.16<x<0.65. Neutron asymmetries were extracted using the effective nucleon polarization and measured proton-to-(3)He cross-section ratios. The measured neutron asymmetries are negative with an average value of (-1.09±0.38)×10(-2) for invariant mass W>2 GeV, which is nonzero at the 2.89σ level. Our measured asymmetry agrees both in sign and magnitude with a two-photon-exchange model prediction that uses input from the Sivers transverse momentum distribution obtained from semi-inclusive deep-inelastic scattering.