A search for the Higgs boson decaying into a photon and a pair of electrons or muons with an invariant mass GeV is presented. The analysis is performed using 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of proton-proton collision data, produced by the LHC at a centre-of-mass energy of 13 TeV and collected by the ATLAS experiment. Evidence for the $H\to \ell \ell \gamma$ process is found with a significance of 3.2$\sigma$ over the background-only hypothesis, compared to an expected significance of 2.1$\sigma$. The best-fit value of the signal strength parameter, defined as the ratio of the observed signal yield to the one expected in the Standard Model, is $\mu =1.5\pm 0.5$. The Higgs boson production cross-section times the $H\to \ell \ell \gamma$ branching ratio for 30 GeV is determined to be 8.7 $\mathrm{<mrow\; data-mjx-texclass="ORD">-2.7</mrow><mrow\; data-mjx-texclass="ORD">+2.8</mrow>}$ fb.