  We report the first measurement of the target-normal single-spin asymmetry in
deep-inelastic scattering from the inclusive reaction
$^3$He$^{\uparrow}\left(e,e' \right)X$ on a polarized $^3$He gas target.
Assuming time-reversal invariance, this asymmetry is strictly zero in the Born
approximation but can be non-zero 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 $(-1.09 \pm 0.38)
\times10^{-2}$ for invariant mass $W>2$ GeV, which is non-zero at the
$2.89\sigma$ 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.
