  We show that the decay rates of the Higgs boson to a pseudoscalar quarkonium
and a pair of leptons, $h\to P\ell^+\ell^-$ ($P\in\{\eta_c,\eta_b\}$), can be
substantially enhanced in a scenario with two Higgs doublets with a softly
broken $\mathbb{Z}_2$ symmetry (2HDM) when the CP-odd Higgs A is light, i.e.
$m_A\lesssim m_h$. Depending on the type of 2HDM the enhancement of
$\mathcal{B}(h\to \eta_{c,b}\tau^+\tau^-)$ with respect to its Standard Model
value can be an order of magnitude larger, i.e.
$\mathcal{O}(10^{-6}\div10^{-5})$. The decays $h\to P\ell^+\ell^-$ could
therefore provide an efficient channel to investigate the presence of a light
CP-odd Higgs $A$ and help to disentangle among various 2HDM scenarios.
