The inclusive J/ψ meson production in Pb–Pb collisions at a center-of-mass energy per nucleon–nucleon collision of s_NN=5.02 TeV at midrapidity (|y|<0.9) is reported by the ALICE Collaboration. The measurements are performed in the dielectron decay channel, as a function of event centrality and J/ψ transverse momentum p_T, down to p_T=0. The J/ψ mean transverse momentum 〈p_T〉 and r_AA ratio, defined as 〈p_T²〉_PbPb/〈p_T²〉_pp, are evaluated. Both observables show a centrality dependence decreasing towards central (head-on) collisions. The J/ψ nuclear modification factor R_AA exhibits a strong p_T dependence with a large suppression at high p_T and an increase to unity for decreasing p_T. When integrating over the measured momentum range p_T<10 GeV/c, the J/ψ R_AA shows a weak centrality dependence. Each measurement is compared with results at lower center-of-mass energies and with ALICE measurements at forward rapidity, as well as to theory calculations. All reported features of the J/ψ production at low p_T are consistent with a dominant contribution to the J/ψ yield originating from charm quark (re)combination.