We report a study of the B meson decays, B+ --> J/psiphiK+, B0 --> J/psiphiK(0)(S), B0 --> J/psiphi, B0 --> J/psieta, and B0 --> J/psieta' using 56 x 10(6) B(-)B events collected at the Upsilon(4S) resonance with the BABAR detector at the PEP-II e(+)e(-) asymmetric-energy storage ring. We measure the branching fractions B(B+ --> J/psiphiK+)=(4.4+/-1.4(stat)+/-0.5(syst))x 10(-5) and B(B0 --> J/psiphiK(0)(S))=(5.1+/-1.9(stat)+/-0.5(syst))x 10(-5), and set upper limits at 90% confidence level for the branching fractions B(B0 --> J/psiphi)<9.2 x 10(-6), B(B0 --> J/psieta)<2.7 x 10(-5), and B(B0 --> J/psieta')<6.3 x 10(-5).