  We present the first theoretical prediction of the production rate of
$1^{-+}$ light hybrid meson $\eta_1$ in $J/\psi$ radiative decays. In the
$N_f=2$ lattice QCD formalism with the pion mass $m_\pi\approx 350$ MeV, the
related electromagnetic multipole form factors are extracted from the
three-point functions that involve necessarily quark annihilation diagrams,
which are calculated through the distillation method. The partial width of
$J/\psi\to \gamma \eta_1$ is determined to be $2.29(77)~\mathrm{eV}$ at the
$\eta_1$ mass $m_{\eta_1}=2.23(4)$ GeV. If $\eta_1$ corresponds to the recently
observed $\eta_1(1855)$ in the process $J/\psi\to \gamma\eta_1(1855)\to \gamma
\eta\eta'$ by BESIII, then the branching fraction $\mathrm{Br}(J/\psi\to
\gamma\eta_1(1855))$ is estimated to be $6.2(2.2)\times 10^{-5}$, which implies
$\mathrm{Br}(\eta_1(1855)\to \eta\eta')\sim 4.3\%$.
