Abstract: A study of B and B-s(0) meson decays into J/psi eta and J/psi eta' final states is performed using a data set of proton-proton collisions at centre-of-mass energies of 7 and 8 TeV, collected by the LCHb experiment and corresponding to 3.0 fb(-1) of integrated luminosity. The decay B-0 -> J/psi eta' is observed for the first time. The following ratios of branching fractions are measured: B(B-0 -> J psi eta')/B(B-s(0) -> J psi eta') = (2.28 +/- 0.65 (stat) +/- 0.010 (syst) +/- 0.13 (f(s)/f(d)) x 10(-2) , B(B-0 -> J psi eta')/B(B-s(0) -> J psi eta') = (1.85 +/- 0.65 (stat) +/- 0.09 (syst) +/- 0.11 (f(s)/f(d)) x 10(-2) where the third uncertainty is related to the present knowledge of f(s)/f(d), the ratio between the probabilities for a b quark to form a B-s(0) or a B-0 meson. The branching fraction ratios are used to determine the parameters of eta-eta' meson mixing. In addition, the first evidence for the decay B-s(0) -> psi(2S)' is reported, and the relative branching fraction is measured, B(B-s(0) -> psi(2S)eta')/B(B-s(0) -> J psi eta') = (38.7 +/- 9.0 (stat) +/- 1.3 (syst) +/- 0.9(B)) x 10(-2), where the third uncertainty is due to the limited knowledge of the branching fractions of J/psi and psi(2S) mesons.