The production of J/ψ is measured at midrapidity (|y| < 0.9) in proton-proton collisions at s = 5.02 and 13 TeV, through the dielectron decay channel, using the ALICE detector at the Large Hadron Collider. The data sets used for the analyses correspond to integrated luminosities of L_int = 19.4 ± 0.4 nb⁻¹ and L_int = 32.2 ± 0.5 nb⁻¹ at s = 5.02 and 13 TeV, respectively. The fraction of non-prompt J/ψ mesons, i.e. those originating from the decay of beauty hadrons, is measured down to a transverse momentum p_T = 2 GeV/c (1 GeV/c) at s = 5.02 TeV (13 TeV). The p_T and rapidity (y) differential cross sections, as well as the corresponding values integrated over p_T and y, are carried out separately for prompt and non-prompt J/ψ mesons. The results are compared with measurements from other experiments and theoretical calculations based on quantum chromodynamics (QCD). The shapes of the p_T and y distributions of beauty quarks predicted by state-of-the-art perturbative QCD models are used to extrapolate an estimate of the b b ¯ pair cross section at midrapidity and in the total phase space. The total b b ¯ cross sections are found to be σ_{bb ¯} = 541 ± 45 (stat.) ± 69 (syst.)−12+10 (extr.) μb and σ_{bb ¯} = 218±37 (stat.)±31 (syst.)−9.1+8.2 (extr.) μb at s = 13 and 5.02 TeV, respectively. The value obtained from the combination of ALICE and LHCb measurements in pp collisions at s = 13 TeV is also provided. [Figure not available: see fulltext.]