The Standard Model predicts several rare Higgs boson decay channels, some of which are currently within experimental reach. Among these is the decay to a lepton pair and a photon, $H\rightarrow\ell\ell\gamma$ ($\ell=e,\mu$). The observation of this decay could open the possibility of studying the CP and coupling properties of the Higgs boson in a complementary way to other analyses. Two distinct regimes with different lepton pair invariant masses are explored: low lepton pair mass ($m_{\ell\ell}<30~\text{GeV}$), where the decay predominantly occurs through the virtual photon, $\gamma^*$, and intermediate lepton pair mass (in a window around $m_{\ell\ell}=91.2~\text{GeV}$), where the decay predominantly occurs through the $Z$ boson. Results are based on $139~\text{fb}^{-1}$ of $pp$ collision data at $13~\text{TeV}$ center-of-mass energy collected with the ATLAS detector at the Large Hadron Collider. An upper limit at 95% confidence level on production cross-section times the branching ratio for $pp\rightarrow H\rightarrow Z\gamma$ is set at 3.6 times the Standard Model prediction (with 2.6 times expected in the presence of the Standard Model Higgs boson). An excess over the background-only hypothesis is observed in the $m_{\ell\ell}<30~\text{GeV}$ regime at $3.2\sigma$ significance ($2.1\sigma$ expected), establishing evidence for the $H\rightarrow\ell\ell\gamma$ decay. The Higgs boson production cross-section times the $H\rightarrow\ell\ell\gamma$ branching ratio for $m_{\ell\ell}<30~\text{GeV}$ is measured to be $8.7^{+2.8}_{-2.7}~\text{fb}$.